Infinite families of 3-designs from a type of five-weight code

Abstract

It has been known for a long time that t-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a t-design. While a lot of progress in the direction of constructing codes from t-designs has been made, only a small amount of work on the construction of t-designs from codes has been done. The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The total number of 2-designs and 3-designs obtained in this paper are exponential in any odd m and the block size of the designs varies in a huge range.

1 Introduction

We start with a brief recall of t-designs. Let ${\mathcal {P}}$ be a set of $v \ge 1$ elements, and let ${\mathcal {B}}$ be a set of k-subsets of ${\mathcal {P}},$ where k is a positive integer with $1 \le k \le v.$ Let t be a positive integer with $t \le k.$ The pair ${\mathbb {D}}= ({\mathcal {P}},\, {\mathcal {B}})$ is called a t-$(v,\, k,\, \lambda )$ design, or simply t-design, if every t-subset of ${\mathcal {P}}$ is contained in exactly $\lambda$ elements of ${\mathcal {B}}.$ The elements of ${\mathcal {P}}$ are called points, and those of ${\mathcal {B}}$ are referred to as blocks. We usually use b to denote the number of blocks in ${\mathcal {B}}.$ A t-design is called simple if ${\mathcal {B}}$ does not contain repeated blocks. In this paper, we consider only simple t-designs. A t-design is called symmetric if $v = b.$ It is clear that t-designs with $k = t$ or $k = v$ always exist. Such t-designs are trivial. In this paper, we consider only t-designs with $v> k > t.$ A t-$(v,\,k,\,\lambda )$ design is referred to as a Steiner system if $t \ge 2$ and $\lambda =1,$ and is denoted by $S(t,\,k,\, v).$

A necessary condition for the existence of a t-$(v,\, k,\, \lambda )$ design is that

$$\begin{aligned} \left( {\begin{array}{c}k-i\\ t-i\end{array}}\right) \text{ divides } \lambda \left( {\begin{array}{c}v-i\\ t-i\end{array}}\right) , \end{aligned}$$

(1)

for all integer i with $0 \le i \le t.$

The interplay between codes and t-designs goes in two directions. In one direction, the incidence matrix of any t-design generates a linear code over any finite field ${\mathrm {GF}}(q).$ A lot of progress in this direction has been made and documented in the literature (see, for examples, [1, 5, 19, 20]). In the other direction, the codewords of a fixed Hamming weight in a linear or nonlinear code may hold a t-design. Some linear and nonlinear codes were employed to construct t-designs [1, 10, 12, 14, 16, 18,19,20]. Binary and ternary Golay codes of certain parameters give 4-designs and 5-designs with fixed parameters. However, the largest t for which an infinite family of t-designs is derived directly from codes is $t=3.$ According to [1, 13, 19, 20], not much progress on the construction of t-designs from codes has been made so far, while many other constructions of t-designs are documented in the literature [3, 4, 13, 15, 17]. The first motivation of this paper is to demonstrate that exponentially many infinite families of 3-designs could be constructed from linear codes. The second motivation is the important applications of t-designs in coding theory, cryptography, communications and statistics.

The objective of this paper is to construct infinite families of 2-designs and 3-designs from a type of binary linear codes with five weights. The obtained t-designs depend only on the weight distribution of the underlying binary codes. The total number of 2-designs and 3-designs presented in this paper are exponential in m,  where $m \ge 5$ is an odd integer. In addition, the block size of the designs can vary in a huge range.

2 The classical construction of t-designs from codes

Let ${\mathcal {C}}$ be a $[v,\, \upkappa ,\, d]$ linear code over ${\mathrm {GF}}(q).$ Let $A_i:=A_i({\mathcal {C}}),$ which denotes the number of codewords with Hamming weight i in ${\mathcal {C}},$ where $0 \le i \le v.$ The sequence $(A_0,\, A_1, \ldots , A_{v})$ is called the weight distribution of ${\mathcal {C}},$ and $\sum _{i=0}^v A_iz^i$ is referred to as the weight enumerator of ${\mathcal {C}}.$ For each k with $A_k \ne 0,$ let ${\mathcal {B}}_k$ denote the set of the supports of all codewords with Hamming weight k in ${\mathcal {C}},$ where the coordinates of a codeword are indexed by $(0,\,1,\,2, \ldots , v-1).$ Let ${\mathcal {P}}=\{0,\, 1,\, 2, \ldots , v-1\}.$ The pair $({\mathcal {P}},\, {\mathcal {B}}_k)$ may be a t-$(v,\, k,\, \lambda )$ design for some positive integer $\lambda .$ The following theorems, developed by Assmus and Mattson, show that the pair $({\mathcal {P}},\, {\mathcal {B}}_k)$ defined by a linear code is a t-design under certain conditions.

Theorem 1

(Assmus–Mattson Theorem [2, 9], p. 303) Let ${\mathcal {C}}$ be a binary $[v,\, \upkappa ,\, d]$ code. Suppose ${\mathcal {C}}^\perp$ has minimum weight $d^\perp .$ Suppose that $A_i=A_i({\mathcal {C}})$ and $A_i^\perp =A_i({\mathcal {C}}^\perp ),$ for $0 \le i \le v,$ are the weight distributions of ${\mathcal {C}}$ and ${\mathcal {C}}^\perp ,$ respectively. Fix a positive integer t with $t < d,$ and let s be the number of i with $A_i^\perp \ne 0$ for $0 < i \le v-t.$ Suppose that $s \le d -t.$ Then

• the codewords of weight i in ${\mathcal {C}}$ hold a t-design provided that $A_i \ne 0$ and $d \le i \le v,$ and

• the codewords of weight i in ${\mathcal {C}}^\perp$ hold a t-design provided that $A_i^\perp \ne 0$ and $d^\perp \le i \le v.$

To construct t-designs via Theorem 1, we will need the following lemma in subsequent sections, which is a variant of the MacWilliam Identity [21, p. 41].

Theorem 2

Let ${\mathcal {C}}$ be a $[v,\, \upkappa ,\, d]$ code over ${\mathrm {GF}}(q)$ with weight enumerator $A(z)=\sum _{i=0}^v A_iz^i$ and let $A^\perp (z)$ be the weight enumerator of ${\mathcal {C}}^\perp .$ Then

$$\begin{aligned}A^\perp (z)=q^{-\upkappa }(1+(q-1)z)^vA\left( \frac{1-z}{1+(q-1)z}\right) .\end{aligned}$$

Later in this paper, we will need also the following theorem.

Theorem 3

Let ${\mathcal {C}}$ be an $[n,\, k,\, d]$ binary linear code, and let ${\mathcal {C}}^\perp$ denote the dual of ${\mathcal {C}}.$ Denote by $\overline{{\mathcal {C}}^\perp }$ the extended code of ${\mathcal {C}}^\perp ,$ and let $\overline{{\mathcal {C}}^\perp }^\perp$ denote the dual of $\overline{{\mathcal {C}}^\perp }.$ Then we have the following.

• (1) ${\mathcal {C}}^\perp$ has parameters $[n,\, n-k,\, d^\perp ],$ where $d^\perp$ denotes the minimum distance of ${\mathcal {C}}^\perp .$

• (2) $\overline{{\mathcal {C}}^\perp }$ has parameters $[n+1,\, n-k,\, \overline{d^\perp }],$ where $\overline{d^\perp }$ denotes the minimum distance of $\overline{{\mathcal {C}}^\perp },$ and is given by

  $$\begin{aligned} \overline{d^\perp } = \left\{ \begin{array}{ll} d^\perp &{} \text{ if } d^\perp \text{ is } \text{ even }, \\ d^\perp + 1 &{} \text{ if } d^\perp \text{ is } \text{ odd }. \end{array} \right. \end{aligned}$$

• (3) $\overline{{\mathcal {C}}^\perp }^\perp$ has parameters $[n+1,\, k+1,\, \overline{d^\perp }^\perp ],$ where $\overline{d^\perp }^\perp$ denotes the minimum distance of $\overline{{\mathcal {C}}^\perp }^\perp .$ Furthermore, $\overline{{\mathcal {C}}^\perp }^\perp$ has only even-weight codewords, and all the nonzero weights in $\overline{{\mathcal {C}}^\perp }^\perp$ are the following:

  $$\begin{aligned} w_1, \, w_2, \ldots , w_t;\, n+1-w_1, \, n+1-w_2, \ldots , n+1-w_t; \, n+1, \end{aligned}$$

  where $w_1,\, w_2,\ldots , w_t$ denote all the nonzero weights of ${\mathcal {C}}.$

Proof

The conclusions of the first two parts are straightforward. We prove only the conclusions of the third part below.

Since $\overline{{\mathcal {C}}^\perp }$ has length $n+1$ and dimension $n-k,$ the dimension of $\overline{{\mathcal {C}}^\perp }^\perp$ is $k+1.$ By assumption, all codes under consideration are binary. By definition, $\overline{{\mathcal {C}}^\perp }$ has only even-weight codewords. Recall that $\overline{{\mathcal {C}}^\perp }$ is the extended code of ${\mathcal {C}}^\perp .$ It is known that the generator matrix of $\overline{{\mathcal {C}}^\perp }^\perp$ is given by [9, p. 15]

$$\begin{aligned} \left[ \begin{array}{cc} {{\bar{\mathbf {1}}}}&{} 1 \\ G &{} {{\bar{\mathbf {0}}}}\end{array} \right] , \end{aligned}$$

where ${{\bar{\mathbf {1}}}}=(1 1 1 \cdots 1)$ is the all-one vector of length n,  ${{\bar{\mathbf {0}}}}=(000 \cdots 0)^T,$ which is a column vector of length n,  and G is the generator matrix of ${\mathcal {C}}.$ Notice again that $\overline{{\mathcal {C}}^\perp }^\perp$ is binary, the desired conclusions on the weights in $\overline{{\mathcal {C}}^\perp }^\perp$ follow from the relation between the two generator matrices of the two codes $\overline{{\mathcal {C}}^\perp }^\perp$ and ${\mathcal {C}}.$ $\square$

3 A type of binary linear codes with five-weights and related codes

In this section, we first introduce a type of binary linear codes ${\mathcal {C}}_m$ of length $n=2^m-1,$ which has the weight distribution of Table 1, and then analyze their dual codes ${\mathcal {C}}_m^\perp ,$ the extended codes $\overline{{\mathcal {C}}_m^\perp },$ and the duals $\overline{{\mathcal {C}}_m^\perp }^\perp .$ Such codes will be employed to construct t-designs in Sects. 4 and 5. Examples of such codes will be given in Sect.  6.

Table 1 The weight distribution of ${\mathcal {C}}_{m}$ for odd m

Theorem 4

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the dual code ${\mathcal {C}}_{m}^\perp$ has parameters $[2^m-1,\, 2^m-1-3m,\, 7],$ and its weight distribution is given by

$$\begin{aligned} 2^{3m}A^\perp _k= & {} \left( {\begin{array}{c}2^m-1\\ k\end{array}}\right) +a U_a(k) + b U_b(k) + c U_c(k) + d U_d(k) + e U_e(k), \end{aligned}$$

where $0 \le k \le 2^m-1$,

$$\begin{aligned} a= & {} \left( 2^m-1\right) 2^{(m-5)/2}\left( 2^{(m-3)/2}+1\right) \left( 2^{m-1}-1\right) /3, \\ b= & {} \left( 2^m-1\right) 2^{(m-3)/2} \left( 2^{(m-1)/2}+1\right) \left( 5\times 2^{m-1}+4\right) /3, \\ c= & {} {\left( 2^m-1\right) }\left( 9\times 2^{2m-4}+3\times 2^{m-3}+1\right) , \\ d= & {} \left( 2^m-1\right) 2^{(m-3)/2} \left( 2^{(m-1)/2}-1\right) \left( 5\times 2^{m-1}+4\right) /3, \\ e= & {} \left( 2^m-1\right) 2^{(m-5)/2} \left( 2^{(m-3)/2}-1\right) \left( 2^{m-1}-1\right) /3, \end{aligned}$$

and

$$\begin{aligned} U_a(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}-2^{(m+1)/2}\\ 0\le j \le 2^{m-1}+2^{(m+1)/2}-1 \\ i+j=k \end{array}}(-1)^i \left( {\begin{array}{c}2^{m-1}-2^{(m+1)/2}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}+2^{(m+1)/2}-1\\ j\end{array}}\right) ,\\ U_b(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}-2^{(m-1)/2}\\ 0\le j \le 2^{m-1}+2^{(m-1)/2}-1 \\ i+j=k \end{array}}(-1)^i \left( {\begin{array}{c}2^{m-1}-2^{(m-1)/2}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}+2^{(m-1)/2}-1\\ j\end{array}}\right) ,\\ U_c(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}\\ 0\le j \le 2^{m-1}-1 \\ i+j=k \end{array}}(-1)^i \left( {\begin{array}{c}2^{m-1}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}-1\\ j\end{array}}\right) ,\\ U_d(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}+2^{(m-1)/2}\\ 0\le j \le 2^{m-1}-2^{(m-1)/2}-1 \\ i+j=k \end{array}}(-1)^i \left( {\begin{array}{c}2^{m-1}+2^{(m-1)/2}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}-2^{(m-1)/2}-1\\ j\end{array}}\right) ,\\ U_e(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}+2^{(m+1)/2}\\ 0\le j \le 2^{m-1}-2^{(m+1)/2}-1 \\ i+j=k \end{array}}(-1)^i \left( {\begin{array}{c}2^{m-1}+2^{(m+1)/2}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}-2^{(m+1)/2}-1\\ j\end{array}}\right) . \end{aligned}$$

Proof

By assumption, the weight enumerator of ${\mathcal {C}}_{m}$ is given by

$$\begin{aligned} A(z)=1+az^{2^{m-1}-2^{(m+1)/2}} +bz^{2^{m-1}-2^{(m-1)/2}} +cz^{2^{m-1}}+dz^{2^{m-1}+2^{(m-1)/2}} +ez^{2^{m-1}+2^{(m+1)/2}}. \end{aligned}$$

It then follows from Theorem 2 that the weight enumerator of ${\mathcal {C}}_{m}^\perp$ is given by

$$\begin{aligned}&2^{3m}A^\perp (z) = (1+z)^{2^m-1}\left[ 1 + a\left( \frac{1-z}{1+z}\right) ^{2^{m-1}-2^{\frac{m+1}{2}}} + b\left( \frac{1-z}{1+z}\right) ^{2^{m-1}-2^{ \frac{m-1}{2} }} \right] \\&\quad + (1+z)^{2^m-1}\left[ c\left( \frac{1-z}{1+z}\right) ^{2^{m-1}}+ d\left( \frac{1-z}{1+z}\right) ^{2^{m-1}+2^{\frac{m-1}{2} }} + e\left( \frac{1-z}{1+z}\right) ^{2^{m-1}+2^{ \frac{m+1}{2} }} \right] . \end{aligned}$$

Hence, we have

$$\begin{aligned} 2^{3m}A^\perp (z)= & {} (1+z)^{2^m-1} \\&\quad + a(1-z)^{2^{m-1}-2^{(m+1)/2}}(1+z)^{2^{m-1}+2^{(m+1)/2}-1} \\&\quad + b(1-z)^{2^{m-1}-2^{(m-1)/2}}(1+z)^{2^{m-1}+2^{(m-1)/2}-1} \\&\quad + c(1-z)^{2^{m-1}}(1+z)^{2^{m-1}-1} \\&\quad + d(1-z)^{2^{m-1}+2^{(m-1)/2}}(1+z)^{2^{m-1}-2^{(m-1)/2}-1} \\&\quad + e(1-z)^{2^{m-1}+2^{(m+1)/2}}(1+z)^{2^{m-1}-2^{(m+1)/2}-1}. \end{aligned}$$

Obviously, we have

$$\begin{aligned} (1+z)^{2^m-1}= & {} \sum _{k=0}^{2^m-1} \left( {\begin{array}{c}2^m-1\\ k\end{array}}\right) z^k. \end{aligned}$$

It is easily seen that

$$\begin{aligned} (1-z)^{2^{m-1}-2^{(m+1)/2}}(1+z)^{2^{m-1}+2^{(m+1)/2}-1} = \sum _{k=0}^{2^m-1} U_a(k) z^k, \end{aligned}$$

and

$$\begin{aligned} (1-z)^{2^{m-1}-2^{(m-1)/2}}(1+z)^{2^{m-1}+2^{(m-1)/2}-1} = \sum _{k=0}^{2^m-1} U_b(k) z^k. \end{aligned}$$

Similarly,

$$\begin{aligned} (1-z)^{2^{m-1}+2^{(m-1)/2}}(1+z)^{2^{m-1}-2^{(m-1)/2}-1} = \sum _{k=0}^{2^m-1} U_d(k) z^k, \end{aligned}$$

and

$$\begin{aligned} (1-z)^{2^{m-1}+2^{(m+1)/2}}(1+z)^{2^{m-1}-2^{(m+1)/2}-1} = \sum _{k=0}^{2^m-1} U_e(k) z^k. \end{aligned}$$

Finally, we have

$$\begin{aligned} (1-z)^{2^{m-1}}(1+z)^{2^{m-1}-1}= \sum _{k=0}^{2^m-1} U_c(k) z^k. \end{aligned}$$

Combining these formulas above yields the weight distribution formula for $A_k^\perp .$

The weight distribution in Table 1 tells us that the dimension of ${\mathcal {C}}_{m}$ is 3m. Therefore, the dimension of ${\mathcal {C}}_{m}^\perp$ is equal to $2^m-1-3m.$ Finally, we prove that the minimum distance of ${\mathcal {C}}_{m}^\perp$ equals 7.

We now prove that $A_k^\perp =0$ for all k with $1 \le k \le 6.$ Let $x=2^{(m-1)/2}.$ With the weight distribution formula for ${\mathcal {C}}_m^\perp$ obtained before, we have

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 1\end{array}}\right)= & {} 2x^2 - 1,\\ a U_a(1)= & {} \frac{1}{3}x^7 + \frac{7}{12}x^6 - \frac{2}{3}x^5 - \frac{7}{8}x^4 + \frac{5}{12}x^3 + \frac{7}{24}x^2 - \frac{1}{12}x, \\ b U_b(1)= & {} \frac{10}{3}x^7 + \frac{5}{3}x^6 - \frac{2}{3}x^5 + \frac{1}{2}x^4 - \frac{11}{6}x^3 - \frac{2}{3}x^2 + \frac{2}{3}x, \\ c U_c(1)= & {} {-}\frac{9}{2}x^6 + \frac{3}{4}x^4 - \frac{5}{4}x^2 + 1, \\ d U_d(1)= & {} {-}\frac{10}{3}x^7 + \frac{5}{3}x^6 + \frac{2}{3}x^5 + \frac{1}{2}x^4 + \frac{11}{6}x^3 - \frac{2}{3}x^2 - \frac{2}{3}x, \\ e U_e(1)= & {} {-}\frac{1}{3}x^7 + \frac{7}{12}x^6 + \frac{2}{3}x^5 - \frac{7}{8}x^4 - \frac{5}{12}x^3 + \frac{7}{24}x^2 + \frac{1}{12}x. \end{aligned}$$

Consequently,

$$\begin{aligned} 2^{3m} A_1^\perp = \left( {\begin{array}{c}2^m-1\\ 1\end{array}}\right) +a U_a(1) + b U_b(1) + c U_c(1) + d U_d(1) + e U_e(1)=0. \end{aligned}$$

Plugging $k=2$ into the weight distribution formula above for ${\mathcal {C}}_m^\perp ,$ we get that

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 2\end{array}}\right)= & {} 2x^4 - 3x^2 + 1,\\ a U_a(2)= & {} \frac{7}{12}x^8 + \frac{5}{6}x^7 - \frac{35}{24}x^6 - \frac{13}{12}x^5 + \frac{7}{6}x^4 + \frac{1}{6}x^3 - \frac{7}{24}x^2 + \frac{1}{12}x,\\ b U_b(2)= & {} \frac{5}{3}x^8 - \frac{5}{3}x^7 - \frac{7}{6}x^6 + \frac{7}{6}x^5 - \frac{7}{6}x^4 + \frac{7}{6}x^3 + \frac{2}{3}x^2 - \frac{2}{3}x, \\ c U_c(2)= & {} {-}\frac{9}{2}x^8 + \frac{21}{4}x^6 - 2x^4 + \frac{9}{4}x^2 - 1, \\ d U_d(2)= & {} \frac{5}{3}x^8 + \frac{5}{3}x^7 - \frac{7}{6}x^6 - \frac{7}{6}x^5 - \frac{7}{6}x^4 - \frac{7}{6}x^3 + \frac{2}{3}x^2 + \frac{2}{3}x, \\ e U_e(2)= & {} \frac{7}{12}x^8 - \frac{5}{6}x^7 - \frac{35}{24}x^6 + \frac{13}{12}x^5 + \frac{7}{6}x^4 - \frac{1}{6}x^3 - \frac{7}{24}x^2 - \frac{1}{12}x. \end{aligned}$$

As a result,

$$\begin{aligned} 2^{3m} A_2^\perp = \left( {\begin{array}{c}2^m-1\\ 2\end{array}}\right) +a U_a(2) + b U_b(2) + c U_c(2) + d U_d(2) + e U_e(2)=0. \end{aligned}$$

Putting $k=3$ into the weight distribution formula above for ${\mathcal {C}}_m^\perp ,$ we obtain that

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 3\end{array}}\right)= & {} \frac{4}{3}x^6 - 4x^4 + \frac{11}{3}x^2 - 1,\\ a U_a(3)= & {} \frac{5}{9}x^9 + \frac{19}{36}x^8 - \frac{14}{9}x^7 + \frac{1}{72}x^6 + \frac{43}{36}x^5 - \frac{17}{18}x^4 - \frac{1}{9}x^3 + \frac{29}{72}x^2 - \frac{1}{12}x,\\ b U_b(3)= & {} {-}\frac{10}{9}x^9 - \frac{25}{9}x^8 + \frac{22}{9}x^7 + \frac{35}{18}x^6 - \frac{7}{18}x^5 + \frac{35}{18}x^4 - \frac{29}{18}x^3 - \frac{10}{9}x^2 + \frac{2}{3}x, \\ c U_c(3)= & {} \frac{9}{2}x^8 - \frac{21}{4}x^6 + 2x^4 - \frac{9}{4}x^2 + 1, \\ d U_d(3)= & {} \frac{10}{9}x^9 - \frac{25}{9}x^8 - \frac{22}{9}x^7 + \frac{35}{18}x^6 + \frac{7}{18}x^5 + \frac{35}{18}x^4 + \frac{29}{18}x^3 - \frac{10}{9}x^2 - \frac{2}{3}x, \\ e U_e(3)= & {} {-}\frac{5}{9}x^9 + \frac{19}{36}x^8 + \frac{14}{9}x^7 + \frac{1}{72}x^6 - \frac{43}{36}x^5 - \frac{17}{18}x^4 + \frac{1}{9}x^3 + \frac{29}{72}x^2 + \frac{1}{12}x. \end{aligned}$$

Hence,

$$\begin{aligned} 2^{3m} A_3^\perp = \left( {\begin{array}{c}2^m-1\\ 3\end{array}}\right) +a U_a(3) + b U_b(3) + c U_c(3) + d U_d(3) + e U_e(3)=0. \end{aligned}$$

Plugging $k=4$ into the weight distribution formula above for ${\mathcal {C}}_m^\perp ,$ we get that

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 4\end{array}}\right)= & {} \frac{2}{3}x^8 - \frac{10}{3}x^6 + \frac{35}{6}x^4 - \frac{25}{6}x^2 + 1,\\ a U_a(4)= & {} \frac{19}{72}x^{10} - \frac{1}{36}x^9 - \frac{25}{48}x^8 + \frac{113}{72}x^7 - \frac{35}{72}x^6 - \frac{77}{36}x^5\\&\quad + \frac{55}{48}x^4 + \frac{37}{72}x^3 - \frac{29}{72}x^2 + \frac{1}{12}x,\\ b U_b(4)= & {} {-}\frac{25}{18}x^{10} - \frac{5}{18}x^9 + \frac{15}{4}x^8 - \frac{53}{36}x^7 - \frac{35}{36}x^6 + \frac{49}{36}x^5 - \frac{5}{2}x^4\\&\quad + \frac{19}{18}x^3 + \frac{10}{9}x^2 - \frac{2}{3}x, \\ c U_c(4)= & {} \frac{9}{4}x^{10} - \frac{57}{8}x^8 + \frac{25}{4}x^6 - \frac{25}{8}x^4 + \frac{11}{4}x^2 - 1, \\ d U_d(4)= & {} {-}\frac{25}{18}x^{10} + \frac{5}{18}x^9 + \frac{15}{4}x^8 + \frac{53}{36}x^7 - \frac{35}{36}x^6 - \frac{49}{36}x^5 - \frac{5}{2}x^4\\&\quad - \frac{19}{18}x^3 + \frac{10}{9}x^2 + \frac{2}{3}x, \\ e U_e(4)= & {} \frac{19}{72}x^{10} + \frac{1}{36}x^9 - \frac{25}{48}x^8 - \frac{113}{72}x^7 - \frac{35}{72}x^6 + \frac{77}{36}x^5\\&\quad + \frac{55}{48}x^4 - \frac{37}{72}x^3 - \frac{29}{72}x^2 - \frac{1}{12}x. \end{aligned}$$

Consequently,

$$\begin{aligned} 2^{3m} A_4^\perp = \left( {\begin{array}{c}2^m-1\\ 4\end{array}}\right) +a U_a(4) + b U_b(4) + c U_c(4) + d U_d(4) + e U_e(4)=0. \end{aligned}$$

Putting $k=5$ into the weight distribution formula above for ${\mathcal {C}}_m^\perp ,$ we obtain that

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 5\end{array}}\right)= & {} \frac{4}{15}x^{10} - 2x^8 + \frac{17}{3}x^6 - \frac{15}{2}x^4 + \frac{137}{30}x^2 - 1,\\ a U_a(5)= & {} {-}\frac{1}{90}x^{11} - \frac{103}{360}x^{10} + \frac{59}{90}x^9 + \frac{1279}{720}x^8 - \frac{97}{40}x^7 - \frac{49}{40}x^6 + \frac{211}{90}x^5\\&\quad - \frac{529}{720}x^4 - \frac{173}{360}x^3 + \frac{169}{360}x^2 - \frac{1}{12}x,\\ b U_b(5)= & {} {-}\frac{1}{9}x^{11} + \frac{23}{18}x^{10} - \frac{14}{45}x^9 - \frac{781}{180}x^8 + \frac{121}{60}x^7 + \frac{91}{60}x^6 - \frac{169}{180}x^5\\&\quad + \frac{263}{90}x^4 - \frac{119}{90}x^3 - \frac{62}{45}x^2 + \frac{2}{3}x, \\ c U_c(5)= & {} {-}\frac{9}{4}x^{10} + \frac{57}{8}x^8 - \frac{25}{4}x^6 + \frac{25}{8}x^4 - \frac{11}{4}x^2 + 1, \\ d U_d(5)= & {} \frac{1}{9}x^{11} + \frac{23}{18}x^{10} + \frac{14}{45}x^9 - \frac{781}{180}x^8 - \frac{121}{60}x^7 + \frac{91}{60}x^6 + \frac{169}{180}x^5\\&\quad + \frac{263}{90}x^4 + \frac{119}{90}x^3 - \frac{62}{45}x^2 - \frac{2}{3}x, \\ e U_e(5)= & {} \frac{1}{90}x^{11} - \frac{103}{360}x^{10} - \frac{59}{90}x^9 + \frac{1279}{720}x^8 + \frac{97}{40}x^7 - \frac{49}{40}x^6 - \frac{211}{90}x^5\\&\quad -\frac{529}{720}x^4 + \frac{173}{360}x^3 + \frac{169}{360}x^2 + \frac{1}{12}x. \end{aligned}$$

Consequently,

$$\begin{aligned} 2^{3m} A_5^\perp = \left( {\begin{array}{c}2^m-1\\ 5\end{array}}\right) +a U_a(5) + b U_b(5) + c U_c(5) + d U_d(5) + e U_e(5)=0. \end{aligned}$$

Plugging $k=6$ into the weight distribution formula above for ${\mathcal {C}}_m^\perp ,$ we arrive at that

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 6\end{array}}\right)= & {} \frac{4}{45}x^{12} - \frac{14}{15}x^{10} + \frac{35}{9}x^8 - \frac{49}{6}x^6 + \frac{406}{45}x^4 - \frac{49}{10}x^2 + 1,\\ a U_a(6)= & {} {-}\frac{103}{1080}x^{12} - \frac{97}{540}x^{11} + \frac{1897}{2160}x^{10} + \frac{571}{1080}x^9 - \frac{1573}{720}x^8 + \frac{193}{120}x^7\\&\quad + \frac{2117}{2160}x^6 - \frac{3061}{1080}x^5 + \frac{385}{432}x^4 + \frac{857}{1080}x^3 - \frac{169}{360}x^2 + \frac{1}{12}x,\\ b U_b(6)= & {} \frac{23}{54}x^{12} + \frac{29}{54}x^{11} - \frac{1471}{540}x^{10} - \frac{613}{540}x^9 + \frac{218}{45}x^8 - \frac{68}{45}x^7 \\&\quad - \frac{293}{540}x^6 + \frac{1033}{540}x^5 - \frac{913}{270}x^4 + \frac{233}{270}x^3 + \frac{62}{45}x^2 - \frac{2}{3}x, \\ c U_c(6)= & {} {-}\frac{3}{4}x^{12} + \frac{37}{8}x^{10} - \frac{221}{24}x^8 + \frac{175}{24}x^6 - \frac{97}{24}x^4 + \frac{37}{12}x^2 - 1, \\ d U_d(6)= & {} \frac{23}{54}x^{12} - \frac{29}{54}x^{11} - \frac{1471}{540}x^{10} + \frac{613}{540}x^9 + \frac{218}{45}x^8 + \frac{68}{45}x^7\\&\quad -\frac{293}{540}x^6 - \frac{1033}{540}x^5 - \frac{913}{270}x^4 - \frac{233}{270}x^3 + \frac{62}{45}x^2 + \frac{2}{3}x, \\ e U_e(6)= & {} {-}\frac{103}{1080}x^{12} + \frac{97}{540}x^{11} + \frac{1897}{2160}x^{10} - \frac{571}{1080}x^9 - \frac{1573}{720}x^8 - \frac{193}{120}x^7\\&\quad + \frac{2117}{2160}x^6 + \frac{3061}{1080}x^5 + \frac{385}{432}x^4 - \frac{857}{1080}x^3 - \frac{169}{360}x^2 - \frac{1}{12}x. \end{aligned}$$

As a result,

$$\begin{aligned} 2^{3m} A_6^\perp = \left( {\begin{array}{c}2^m-1\\ 6\end{array}}\right) +a U_a(6) + b U_b(6) + c U_c(6) + d U_d(6) + e U_e(6)=0. \end{aligned}$$

Plugging $k=7$ into the weight distribution formula above for ${\mathcal {C}}_m^\perp ,$ we obtain

$$\begin{aligned} \left( {\begin{array}{c}2^m-1\\ 7\end{array}}\right)= & {} \frac{8}{315}x^{14} - \frac{16}{45}x^{12} + \frac{92}{45}x^{10} - \frac{56}{9}x^8 + \frac{967}{90}x^6 - \frac{469}{45}x^4 + \frac{363}{70}x^2 - 1, \end{aligned}$$

and

$$\begin{aligned} a U_a(7)= & {} - \frac{97}{1890}x^{13} - \frac{11}{1512}x^{12} + \frac{125}{378}x^{11} - \frac{8711}{15,120}x^{10} - \frac{523}{7560}x^9 + \frac{15,643}{5040}x^8\\&- \frac{18,281}{7560}x^7 {-} \frac{39,307}{15,120}x^6 {+} \frac{23,141}{7560}x^5 {-} \frac{6619}{15,120}x^4 {-} \frac{5818}{7560}x^3 + \frac{1303}{2520}x^2 - \frac{1}{12}x, \\ b U_b(7)= & {} \frac{29}{189} x^{13} - \frac{103}{378} x^{12} - \frac{814}{945} x^{11} + \frac{9071}{3780} x^{10} + \frac{2659}{3780} x^9 - \frac{554}{105} x^8 + \frac{3889}{1890} x^7\\&\quad + \frac{4117}{3780} x^6 - \frac{6299}{3780} x^5 + \frac{6857}{1890} x^4 - \frac{1991}{1890} x^3 - \frac{494}{315} x^2 + \frac{2}{3} x, \\ c U_c(7)= & {} \frac{3}{4} x^{12} - \frac{37}{8} x^{10} + \frac{221}{24} x^8 - \frac{175}{24}x^6 + \frac{97}{24} x^4 - \frac{37}{12} x^2 + 1, \\ d U_d(7)= & {} - \frac{29}{189} x^{13} - \frac{103}{378} x^{12} + \frac{814}{945} x^{11} + \frac{9071}{3780} x^{10} - \frac{2659}{3780} x^9 - \frac{554}{105} x^8 - \frac{3889}{1890} x^7\\&\quad + \frac{4117}{3780} x^6 {+} \frac{6299}{3780} x^5 {+} \frac{6857}{1890} x^4 {+} \frac{1991}{1890} x^3 {-} \frac{494}{315} x^2 {-} \frac{2}{3} x, \\ e U_e(7)= & {} \frac{97}{1890} x^{13} {-} \frac{11}{1512} x^{12} {-} \frac{125}{378} x^{11} {-} \frac{8711}{15,120} x^{10} {+} \frac{523}{7560} x^9 {+} \frac{15,643}{5040} x^8 {+} \frac{18,281}{7560} x^7\\&\quad - \frac{39,307}{15,120} x^6 - \frac{23,141}{7560} x^5 - \frac{6619}{15,120} x^4 + \frac{5819}{7560} x^3 + \frac{1303}{2520} x^2 + \frac{1}{12} x. \end{aligned}$$

It then follows that

$$\begin{aligned} A_7^\perp= & {} 2^{-3m} \left( \left( {\begin{array}{c}2^m-1\\ 7\end{array}}\right) +a U_a(7) + b U_b(7) + c U_c(7) + d U_d(7) + e U_e(7)\right) \\= & {} \frac{(x^2-1) (2x^2 - 1) (x^4 - 5x^2 + 34)}{630} . \end{aligned}$$

Notice that $x^4 - 5x^2 + 34=(x^2-5/2)^2 +34-25/4 >0.$ We have $A_7^\perp >0$ for all odd $m \ge 5.$ This proves the desired conclusion on the minimum distance of ${\mathcal {C}}_{m}^\perp .$ $\square$

Theorem 5

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. The code $\overline{{\mathcal {C}}_{m}^\perp }^\perp$ has parameters

$$\begin{aligned} \left[ 2^m, \, 3m+1, \, 2^{m-1}-2^{(m+1)/2}\right] , \end{aligned}$$

and its weight enumerator is given by

$$\begin{aligned}&\overline{A^\perp }^\perp (z) = 1+uz^{2^{m-1}-2^{\frac{m+1}{2} }} + vz^{2^{m-1}-2^{ \frac{m-1}{2}}} \nonumber \\&\quad + wz^{2^{m-1}} + vz^{2^{m-1}+2^{ \frac{m-1}{2}}} + uz^{2^{m-1}+2^{\frac{m+1}{2} }} +z^{2^m}, \end{aligned}$$

(2)

where

$$\begin{aligned} u= & {} \frac{2^{3m-4} - 3 \times 2^{2m-4} + 2^{m-3}}{3}, \\ v= & {} \frac{5\times 2^{3m-2} + 3 \times 2^{2m-2} - 2^{m+1}}{3} , \\ w= & {} {2\left( 2^m-1\right) } \left( 9\times 2^{2m-4}+3\times 2^{m-3}+1\right) . \end{aligned}$$

Proof

It follows from Theorem 3 that the code has all the weights given in (2). It remains to determine the frequencies of these weights. The weight distribution of the code ${\mathcal {C}}_{m}$ given in Table 1 and the generator matrix of the code $\overline{{\mathcal {C}}_{m}^\perp }^\perp$ documented in the proof of Theorem 3 show that

$$\begin{aligned} \overline{A^\perp }^\perp _{2^{m-1}}=2c=w, \end{aligned}$$

where c was defined in Theorem 4.

We now determine u and v. Recall that ${\mathcal {C}}_{m}^\perp$ has minimum distance 7. It then follows from Theorem 3 that $\overline{{\mathcal {C}}_{m}^\perp }$ has minimum distance 8. The first and third Pless power moments say that

$$\begin{aligned} \left\{ \begin{array}{l} \sum _{i=0}^{2^m} \overline{A^\perp }^\perp _{i} = 2^{3m+1}, \\ \sum _{i=0}^{2^m} i^2 \overline{A^\perp }^\perp _{i} = 2^{3m-1} 2^m(2^m+1). \end{array} \right. \end{aligned}$$

These two equations become

$$\begin{aligned} \left\{ \begin{array}{l} 1+u+v+c = 2^{3m}, \\ (2^{2m-2} +2^{m+1}) u + (2^{2m-2} +2^{m-1}) v + 2^{2m-2}c + 2^{2m-1} = 2^{4m-2}(2^m+1). \end{array} \right. \end{aligned}$$

Solving this system of equations proves the desired conclusion on the weight enumerator of this code. $\square$

Finally, we settle the weight distribution of the code $\overline{{\mathcal {C}}_{m}^\perp }.$

Theorem 6

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. The code $\overline{{\mathcal {C}}_{m}^\perp }$ has parameters $[2^m,\, 2^m-1-3m,\, 8],$ and its weight distribution is given by

$$\begin{aligned} 2^{3m+1}\overline{A^\perp }_k = \left( 1+(-1)^k \right) \left( {\begin{array}{c}2^m\\ k\end{array}}\right) + w E_0(k) + u E_1(k) + v E_2(k), \end{aligned}$$

(3)

where $w,\, u,\, v$ are defined in Theorem 5, and

$$\begin{aligned} E_0(k)= & {} \frac{1+(-1)^k}{2} (-1)^{\lfloor k/2 \rfloor } \left( {\begin{array}{c}2^{m-1}\\ \lfloor k/2 \rfloor \end{array}}\right) ,\\ E_1(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}-2^{(m+1)/2} \\ 0\le j \le 2^{m-1}+2^{(m+1)/2} \\ i+j=k \end{array}} [(-1)^i +(-1)^j] \left( {\begin{array}{c}2^{m-1}-2^{(m+1)/2}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}+2^{(m+1)/2}\\ j\end{array}}\right) ,\\ E_2(k)= & {} \sum _{\begin{array}{c} 0 \le i \le 2^{m-1}-2^{(m-1)/2} \\ 0\le j \le 2^{m-1}+2^{(m-1)/2} \\ i+j=k \end{array}} [(-1)^i +(-1)^j] \left( {\begin{array}{c}2^{m-1}-2^{(m-1)/2}\\ i\end{array}}\right) \left( {\begin{array}{c}2^{m-1}+2^{(m-1)/2}\\ j\end{array}}\right) , \end{aligned}$$

where $0 \le k \le 2^m.$

Proof

By definition,

$$\begin{aligned} \dim \left( \overline{{\mathcal {C}}_{m}^\perp } \right) = \dim \left( {\mathcal {C}}_{m}^\perp \right) =2^m-1-3m. \end{aligned}$$

It has been showed in the proof of Theorem 4 that the minimum distance of $\overline{{\mathcal {C}}_{m}^\perp }$ is equal to 8. We now prove the conclusion on the weight distribution of this code.

By Theorems 2 and 5, the weight enumerator of $\overline{{\mathcal {C}}_{m}^\perp }$ is given by

$$\begin{aligned} 2^{3m+1}\overline{A^\perp }(z)= & {} (1+z)^{2^m}\left[ 1 + \left( \frac{1-z}{1+z}\right) ^{2^m} + w\left( \frac{1-z}{1+z}\right) ^{2^{m-1}} \right] \nonumber \\&\quad + (1+z)^{2^m}\left[ u\left( \frac{1-z}{1+z}\right) ^{2^{m-1}-2^{\frac{m+1}{2} }}+ v \left( \frac{1-z}{1+z}\right) ^{2^{m-1}-2^{\frac{m-1}{2} }} \right] \nonumber \\&\quad + (1+z)^{2^m}\left[ v \left( \frac{1-z}{1+z}\right) ^{2^{m-1}+2^{\frac{m-1}{2} }} + u\left( \frac{1-z}{1+z}\right) ^{2^{m-1}+2^{\frac{m+1}{2}}} \right] . \end{aligned}$$

(4)

Consequently, we have

$$\begin{aligned} 2^{3m+1}\overline{A^\perp }(z)= & {} (1+z)^{2^m} + (1-z)^{2^m} + w (1-z^2)^{2^{m-1}} \nonumber \\&\quad + u(1-z)^{2^{m-1}-2^{(m+1)/2}}(1+z)^{2^{m-1}+2^{(m+1)/2}} \nonumber \\&\quad + v(1-z)^{2^{m-1}-2^{(m-1)/2}}(1+z)^{2^{m-1}+2^{(m-1)/2}} \nonumber \\&\quad + v(1-z)^{2^{m-1}+2^{(m-1)/2}}(1+z)^{2^{m-1}-2^{(m-1)/2}} \nonumber \\&\quad + u(1-z)^{2^{m-1}+2^{(m+1)/2}}(1+z)^{2^{m-1}-2^{(m+1)/2}}. \end{aligned}$$

(5)

We now treat the terms in (5) one by one. We first have

$$\begin{aligned} (1+z)^{2^m} + (1-z)^{2^m} = \sum _{k=0}^{2^m} \left( 1+(-1)^k \right) \left( {\begin{array}{c}2^m\\ k\end{array}}\right) . \end{aligned}$$

(6)

One can easily see that

$$\begin{aligned} \left( 1-z^2\right) ^{2^{m-1}} = \sum _{i=0}^{2^{m-1}} (-1)^i \left( {\begin{array}{c}2^{m-1}\\ i\end{array}}\right) z^{2i} = \sum _{k=0}^{2^{m}} \frac{1+(-1)^k}{2} (-1)^{\lfloor k/2 \rfloor } \left( {\begin{array}{c}2^{m-1}\\ \lfloor k/2 \rfloor \end{array}}\right) z^{k}. \end{aligned}$$

(7)

Notice that

$$\begin{aligned} (1-z)^{2^{m-1}-2^{(m+1)/2}}=\sum _{i=0}^{2^{m-1}-2^{(m+1)/2}} \left( {\begin{array}{c}2^{m-1}-2^{(m+1)/2}\\ i\end{array}}\right) (-1)^i z^i, \end{aligned}$$

and

$$\begin{aligned} (1+z)^{2^{m-1}+2^{(m+1)/2}}=\sum _{i=0}^{2^{m-1}+2^{(m+1)/2}} \left( {\begin{array}{c}2^{m-1}+2^{(m+1)/2}\\ i\end{array}}\right) z^i. \end{aligned}$$

We have then

$$\begin{aligned} (1-z)^{2^{m-1}-2^{(m+1)/2}} (1+z)^{2^{m-1}+2^{(m+1)/2}} = \sum _{k=0}^{2^m} E_1(k) z^k. \end{aligned}$$

(8)

Similarly, we have

$$\begin{aligned} (1-z)^{2^{m-1}-2^{(m-1)/2}} (1+z)^{2^{m-1}+2^{(m-1)/2}}= & {} \sum _{k=0}^{2^m} E_2(k) z^k, \end{aligned}$$

(9)

$$\begin{aligned} (1-z)^{2^{m-1}+2^{(m-1)/2}} (1+z)^{2^{m-1}-2^{(m-1)/2}}= & {} \sum _{k=0}^{2^m} E_3(k) z^k, \end{aligned}$$

(10)

$$\begin{aligned} (1-z)^{2^{m-1}+2^{(m+1)/2}} (1+z)^{2^{m-1}-2^{(m+1)/2}}= & {} \sum _{k=0}^{2^m} E_4(k) z^k. \end{aligned}$$

(11)

Plugging (6)–(11) into (5) proves the desired conclusion. $\square$

4 Infinite families of 2-designs from ${\mathcal {C}}_{m}^\perp$ and ${\mathcal {C}}_{m}$

Theorem 7

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Let ${\mathcal {P}}=\{0,\,1,\,2, \ldots , 2^m-2\},$ and let ${\mathcal {B}}$ be the set of the supports of the codewords of ${\mathcal {C}}_{m}$ with weight k,  where $A_k \ne 0.$ Then $({\mathcal {P}},\, {\mathcal {B}})$ is a 2-$(2^m-1,\, k,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{k(k-1)A_k}{(2^m-1)(2^m-2)}, \end{aligned}$$

where $A_k$ is given in Table 1.

Let ${\mathcal {P}}=\{0,\,1,\,2, \ldots , 2^m-2\},$ and let ${\mathcal {B}}^\perp$ be the set of the supports of the codewords of ${\mathcal {C}}_{m}^\perp$ with weight k and $A_k^\perp \ne 0.$ Then $({\mathcal {P}},\, {\mathcal {B}}^\perp )$ is a 2-$(2^m-1,\, k,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{k(k-1)A_k^\perp }{(2^m-1)(2^m-2)}, \end{aligned}$$

where $A_k^\perp$ is given in Theorem 4.

Proof

The weight distribution of ${\mathcal {C}}_{m}^\perp$ is given in Theorem 4 and that of ${\mathcal {C}}_{m}$ is given in Table 1. By Theorem 4, the minimum distance $d^\perp$ of ${\mathcal {C}}_{m}^\perp$ is equal to 7. Put $t=2.$ The number of i with $A_i \ne 0$ and $1 \le i \le 2^m-1 -t$ is $s=5.$ Hence, $s=d^\perp -t.$ The desired conclusions then follow from Theorem 1 and the fact that two binary vectors have the same support if and only if they are equal. $\square$

Example 1

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the BCH code ${\mathcal {C}}_{m}$ holds five 2-designs with the following parameters:

• $(v,\, k, \, \lambda )=\left( 2^m-1,\ 2^{m-1}-2^{\frac{m+1}{2}}, \ \frac{2^{\frac{m-5}{2}} \left( 2^{\frac{m-3}{2}}+1\right) \left( 2^{m-1} - 2^{\frac{m+1}{2}} \right) \left( 2^{m-1} - 2^{\frac{m+1}{2}} -1\right) }{6} \right) .$

• $(v,\, k, \, \lambda )=\left( 2^m-1,\ 2^{m-1}-2^{\frac{m-1}{2}}, \ \frac{2^{m-2} \left( 2^{m-1} - 2^{\frac{m-1}{2} } -1 \right) ( 5 \times 2^{m-1} + 4 )}{6} \right) .$

• $(v, \, k, \, \lambda )=\left( 2^m-1, \ 2^{m-1}, \ 2^{m-2} \left( 9 \times 2^{2m-4}+ 3 \times 2^{m-3} +1\right) \right) .$

• $(v,\, k, \, \lambda )=\left( 2^m-1,\ 2^{m-1}+2^{\frac{m-1}{2}}, \ \frac{2^{m-2} \left( 2^{m-1} + 2^{\frac{m-1}{2} } -1 \right) ( 5 \times 2^{m-1} + 4 )}{6} \right) .$

• $(v,\, k, \, \lambda )=\left( 2^m-1,\ 2^{m-1}+2^{\frac{m+1}{2}}, \ \frac{2^{\frac{m-5}{2}} \left( 2^{\frac{m-3}{2}}-1\right) \left( 2^{m-1} + 2^{\frac{m+1}{2}} \right) \left( 2^{m-1} + 2^{\frac{m+1}{2}} -1\right) }{6} \right) .$

Example 2

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the supports of all codewords of weight 7 in ${\mathcal {C}}_{m}^\perp$ give a 2-$(2^m-1,\, 7,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{ 2^{2(m-1)} - 5 \times 2^{m-1} + 34 }{30}. \end{aligned}$$

Proof

By Theorem 4, we have

$$\begin{aligned} A^\perp _7=\frac{(2^{m-1}-1) (2^m-1) (2^{2(m-1)} - 5 \times 2^{m-1} + 34)}{630}. \end{aligned}$$

The desired conclusion on $\lambda$ follows from Theorem 7. $\square$

Example 3

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the supports of all codewords of weight 8 in ${\mathcal {C}}_{m}^\perp$ give a 2-$(2^m-1,\, 8,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{ (2^{m-1}-4)(2^{2(m-1)} - 5 \times 2^{m-1} + 34) }{90}. \end{aligned}$$

Proof

By Theorem 4, we have

$$\begin{aligned} A^\perp _8=\frac{(2^{m-1}-1) (2^{m-1}-4) (2^m-1) (2^{2(m-1)} - 5 \times 2^{m-1} + 34)}{2520}. \end{aligned}$$

The desired conclusion on $\lambda$ follows from Theorem 7. $\square$

Example 4

Let $m \ge 7$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the supports of all codewords of weight 9 in ${\mathcal {C}}_{m}^\perp$ give a 2-$(2^m-1,\, 9,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{ (2^{m-1}-4)(2^{m-1}-16)(2^{2(m-1)} - 2^{m-1} + 28) }{315}. \end{aligned}$$

Proof

By Theorem 4, we have

$$\begin{aligned} A^\perp _9=\frac{(2^{m-1}-1) (2^{m-1}-4) (2^{m-1}-16) (2^m-1) (2^{2(m-1)} - 2^{m-1} + 28)}{11{,}340}. \end{aligned}$$

The desired conclusion on $\lambda$ follows from Theorem 7. $\square$

5 Infinite families of 3-designs from $\overline{{\mathcal {C}}_{m}^\perp }$ and $\overline{{\mathcal {C}}_{m}^\perp }^\perp$

Theorem 8

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Let ${\mathcal {P}}=\{0,\,1,\,2, \ldots , 2^m-1\},$ and let $\overline{{\mathcal {B}}^\perp }^\perp$ be the set of the supports of the codewords of $\overline{{\mathcal {C}}_{m}^\perp }^\perp$ with weight k,  where $\overline{A^\perp }^\perp _k \ne 0.$ Then $({\mathcal {P}},\, \overline{{\mathcal {B}}^\perp }^\perp )$ is a 3-$(2^m,\, k,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{\overline{A^\perp }^\perp _k\left( {\begin{array}{c}k\\ 3\end{array}}\right) }{\left( {\begin{array}{c}2^m\\ 3\end{array}}\right) }, \end{aligned}$$

where $\overline{A^\perp }^\perp _k$ is given in Theorem 5.

Let ${\mathcal {P}}=\{0,\,1,\,2, \ldots , 2^m-1\},$ and let $\overline{{\mathcal {B}}^\perp }$ be the set of the supports of the codewords of $\overline{{\mathcal {C}}_{m}^\perp }$ with weight k and $\overline{A^\perp }_k \ne 0.$ Then $({\mathcal {P}},\, \overline{{\mathcal {B}}^\perp })$ is a 3-$(2^m,\, k,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{\overline{A^\perp }_k \left( {\begin{array}{c}k\\ 3\end{array}}\right) }{\left( {\begin{array}{c}2^m\\ 3\end{array}}\right) }, \end{aligned}$$

where $\overline{A^\perp }_k$ is given in Theorem 6.

Proof

The weight distributions of $\overline{{\mathcal {C}}_{m}^\perp }^\perp$ and $\overline{{\mathcal {C}}_{m}^\perp }$ are described in Theorems 5 and 6. Notice that the minimum distance $\overline{d^\perp }$ of $\overline{{\mathcal {C}}_{m}^\perp }^\perp$ is equal to 8. Put $t=3.$ The number of i with $\overline{A^\perp }_i \ne 0$ and $1 \le i \le 2^m -t$ is $s=5.$ Hence, $s=\overline{d^\perp }-t.$ Clearly, two binary vectors have the same support if and only if they are equal. The desired conclusions then follow from Theorem 1. $\square$

Example 5

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then $\overline{{\mathcal {C}}_{m}^\perp }^\perp$ holds five 3-designs with the following parameters:

• $(v,\, k, \, \lambda )=\left( 2^m,\ 2^{m-1}-2^{\frac{m+1}{2}}, \ \frac{ \left( 2^{m-1} - 2^{\frac{m+1}{2}} \right) \left( 2^{m-1} - 2^{\frac{m+1}{2}} -1\right) \left( 2^{m-1} - 2^{\frac{m+1}{2}} -2\right) }{48} \right) .$

• $(v,\, k, \, \lambda )=\left( 2^m,\ 2^{m-1}-2^{\frac{m-1}{2}}, \ \frac{ 2^{\frac{m-1}{2} } \left( 2^{m-1} - 2^{\frac{m-1}{2} } -1 \right) \left( 2^{\frac{m-1}{2} } -2 \right) ( 5 \times 2^{m-3} + 1 )}{3} \right) .$

• $(v, \, k, \, \lambda )=\left( 2^m, \ 2^{m-1}, \ \left( 2^{m-2}-1\right) \left( 9 \times 2^{2m-4}+ 3 \times 2^{m-3} +1\right) \right) .$

• $(v,\, k, \, \lambda )=\left( 2^m,\ 2^{m-1}+2^{\frac{m-1}{2}}, \ \frac{ 2^{\frac{m-1}{2} } \left( 2^{m-1} + 2^{\frac{m-1}{2} } -1 \right) \left( 2^{\frac{m-1}{2} } +2 \right) ( 5 \times 2^{m-3} + 1 )}{3} \right) .$

• $(v,\, k, \, \lambda )=\left( 2^m,\ 2^{m-1}+2^{\frac{m+1}{2}}, \ \frac{ \left( 2^{m-1} + 2^{\frac{m+1}{2}} \right) \left( 2^{m-1} + 2^{\frac{m+1}{2}} -1\right) \left( 2^{m-1} + 2^{\frac{m+1}{2}} -2\right) }{48} \right) .$

Example 6

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the supports of all codewords of weight 8 in $\overline{{\mathcal {C}}_{m}^\perp }$ give a 3-$(2^m,\, 8,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{2^{2(m-1)} - 5 \times 2^{m-1}+34}{30}. \end{aligned}$$

Proof

By Theorem 6, we have

$$\begin{aligned} \overline{A^\perp }_8=\frac{2^m(2^{m-1}-1)(2^m-1)(2^{2(m-1)} - 5 \times 2^{m-1}+34)}{315}. \end{aligned}$$

The desired value of $\lambda$ follows from Theorem 8. $\square$

Example 7

Let $m \ge 7$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the supports of all codewords of weight 10 in $\overline{{\mathcal {C}}_{m}^\perp }$ give a 3-$(2^m,\, 10,\, \lambda )$ design, where

$$\begin{aligned} \lambda =\frac{(2^{m-1}-4) (2^{m-1}-16) (2^{2(m-1)}-2^{m-1}+28)}{315}. \end{aligned}$$

Proof

By Theorem 6, we have

$$\begin{aligned} \overline{A^\perp }_{10}=\frac{2^{m-1}(2^{m-1}-1)(2^m-1)(2^{m-1}-4) (2^{m-1}-16) (2^{2(m-1)}-2^{m-1}+28) }{4 \times 14{,}175}. \end{aligned}$$

The desired value of $\lambda$ follows from Theorem 8. $\square$

Example 8

Let $m \ge 5$ be an odd integer and let ${\mathcal {C}}_m$ be a binary code with the weight distribution of Table 1. Then the supports of all codewords of weight 12 in $\overline{{\mathcal {C}}_{m}^\perp }$ give a 3-$(2^m,\, 12,\, \lambda )$ design, where

$$\begin{aligned} \lambda {=}\frac{(2^{h-2}-1) (2 \times 2^{5h} {-} 55 \times 2^{4h} {+} 647 \times 2^{3h} - 2727 \times 2^{2h} {+} 11{,}541 \times 2^{h} - 47,208)}{2835}, \end{aligned}$$

and $h=m-1.$

Proof

By Theorem 6, we have

$$\begin{aligned} \overline{A^\perp }_{12}{=}\frac{\epsilon ^2 (\epsilon ^2-1) (\epsilon ^2-4) (2 \epsilon ^2-1) (2 \epsilon ^{10} {-} 55 \epsilon ^8 {+} 647 \epsilon ^6 {-} 2727 \epsilon ^4 {+} 11,541 \epsilon ^2 - 47,208) }{8 \times 467{,}775}, \end{aligned}$$

where $\epsilon =2^{(m-1)/2}.$ The desired value of $\lambda$ follows from Theorem 8. $\square$

6 Two families of binary cyclic codes with the weight distribution of Table 1

To prove the existence of the 2-designs in Sect.  4 and the 3-designs in Sect.  5, we present two families of binary codes of length $2^m-1$ with the weight distribution of Table 1.

Let $n=q^m-1,$ where m is a positive integer. Let $\alpha$ be a generator of ${\mathrm {GF}}(q^m)^*.$ For any i with $0 \le i \le n-1,$ let $\mathbb {M}_i(x)$ denote the minimal polynomial of $\beta ^i$ over ${\mathrm {GF}}(q).$ For any $2 \le \updelta \le n,$ define

$$\begin{aligned} g_{(q,n,\updelta ,b)}(x)={\mathrm {lcm}}\left( \mathbb {M}_{b}(x), \,\mathbb {M}_{b+1}(x), \ldots , \mathbb {M}_{b+\updelta -2}(x)\right) , \end{aligned}$$

(12)

where b is an integer, ${\mathrm {lcm}}$ denotes the least common multiple of these minimal polynomials, and the addition in the subscript $b+i$ of $\mathbb {M}_{b+i}(x)$ always means the integer addition modulo n. Let ${\mathcal {C}}_{(q, n, \updelta ,b)}$ denote the cyclic code of length n with generator polynomial $g_{(q, n,\updelta , b)}(x).\,{\mathcal {C}}_{(q, n, \updelta , b)}$ is called a primitive BCH code with designed distance $\updelta .$ When $b=1,$ the set ${\mathcal {C}}_{(q, n, \updelta , b)}$ is called a narrow-sense primitive BCH code.

Although primitive BCH codes are not asymptotically good, they are among the best linear codes when the length of the codes is not very large [5, Appendix A]. So far, we have very limited knowledge of BCH codes, as the dimension and minimum distance of BCH codes are in general open, in spite of some recent progress [6, 7]. However, in a few cases the weight distribution of a BCH code can be settled. The following theorem introduces such a case.

Theorem 9

Let $m \ge 5$ be an odd integer and let $\updelta =2^{m-1}-1-2^{(m+1)/2}.$ Then the BCH code ${\mathcal {C}}_{(2, 2^m-1, \updelta , 0)}$ has length $n=2^m-1,$ dimension 3m,  and the weight distribution in Table 1.

Proof

A proof can be found in [8]. $\square$

It is known that the dual of a BCH code may not be a BCH code. The following theorem describes a family of cyclic codes having the weight distribution of Table 1, which may not be BCH codes.

Theorem 10

Let $m \ge 5$ be an odd integer. Let ${\mathcal {C}}_m$ be the dual of the narrow-sense primitive BCH code ${\mathcal {C}}_{(2, 2^m-1, 7, 1)}.$ Then ${\mathcal {C}}_m$ has the weight distribution of Table 1.

Proof

A proof can be found in [11]. $\square$

7 Summary and concluding remarks

In this paper, with any binary linear code of length $2^m-1$ and the weight distribution of Table 1, exponentially many infinite families of 2-designs and 3-designs with various block sizes were constructed with only one strike. These designs depend only on the weight distribution of the underlying linear code ${\mathcal {C}}_m,$ and do not depend on the specific construction of the linear code ${\mathcal {C}}_m.$ In other words, one can tell you that your code and its associated codes (the dual code, the extended code of the dual code) hold exponentially many 2-designs and 3-designs if you only tell him/her that you have a binary linear code with the weight distribution of Table 1 without giving further information of your linear code. This fact makes Theorems  7 and 8 different from theorems on t-designs from codes documented in the literature, which need the description of the specific construction of the underlying code. In summary, Theorems 7 and 8 are more specific than the original Assmus–Mattson Theorem, as they work only for a type of linear codes with five weights. They are more general than other theorems on t-designs, as most theorems on t-designs in the literature apply only to a specific linear code.

Given only the weight distribution of a linear code, it might be impossible to determine the automorphism group of the linear code. Thus, Theorems 7 and 8 may not be proved with the automorphism group approach. Therefore, the proofs of 7 and 8 given in the paper may be the only choice. For the same reason, the proofs of Theorems 4 and 6 presented in this paper may not have a choice, though they are complicated and tedious.

The constructions of the exponentially many infinite families of 3-designs presented in this paper demonstrate that the coding theory approach to constructing t-designs may be promising, and may stimulate further investigations in this direction. However, it is open if the codewords of a fixed weight in a family of linear codes can hold an infinite family of t-designs for some $t \ge 4.$