A note on the generalized Hamming weights of Reed–Muller codes

Abstract

In this note, we give a very simple description of the generalized Hamming weights of Reed–Muller codes. For this purpose, we generalize the well-known Macaulay representation of a nonnegative integer and state some of its basic properties.

1 Preliminaries

Let $\mathbb {F}_q$ be the finite field with q elements and denote by ${\mathbb A}^m:={\mathbb A}^m(\mathbb {F}_q)$ the m-dimensional affine space defined over $\mathbb {F}_q$. This space consists of $q^m$ points $(a_1,\ldots ,a_m)$ with $a_1,\ldots ,a_m \in \mathbb {F}_q$. Let $T(m):=\mathbb {F}_q[x_1, \ldots , x_m]$ denote the ring of polynomials in m variables and coefficients in $\mathbb {F}_q.$ Further let $T_{\le d}(m)$ be the set of polynomials in T(m) of total degree at most d. A monomial $X_1^{\alpha _1}\cdots X_m^{\alpha _m}$ is called reduced if $(\alpha _1,\ldots ,\alpha _m) \in \{0,1,\ldots ,q-1\}^m$. Similarly a polynomial $f \in T(m)$ is called reduced if it is an $\mathbb {F}_q$-linear combination of reduced monomials. We denote the set of reduced polynomials by $T^\mathrm {red}(m)$ and define $T^\mathrm {red}_{\le d}(m):=T_{\le d}(m) \cap T^{\mathrm {red}}(m)$.

One reason for considering reduced polynomials comes from coding theory. Indeed, Reed–Muller codes are obtained by evaluating certain polynomials in the points of ${\mathbb A}^m$, but the evaluation map

$$\begin{aligned} \mathrm {Ev}: T(m) \rightarrow \mathbb {F}_q^{q^m}, \quad \hbox {defined by} \ \mathrm {Ev}(f)=(f(P))_{P \in {\mathbb A}^m} \end{aligned}$$

is not injective. However, its restriction to $T^\mathrm {red}(m)$ is. In fact, the kernel of $\mathrm {Ev}$ consists precisely of the ideal $I \subset T(m)$ generated by the polynomials $x_i^q-x_i$ ($1 \le i \le m$). Working with reduced polynomials is simply a convenient way to take this into account, since for two reduced polynomials $f_1,f_2 \in T(m)$ the equality $f_1+I=f_2+I$ holds if and only if $f_1=f_2$.

The Reed–Muller code $\mathrm {RM}_q(d,m)$ is the set of vectors from $\mathbb {F}_q^{q^m}$ obtained by evaluating polynomials of total degree up to d in the $q^m$ points of ${\mathbb A}^m$, that is to say:

$$\begin{aligned} \mathrm {RM}_q(d,m):=\{(f(P))_{P \in {\mathbb A}^m} \, : \, f \in T_{\le d}(m)\}. \end{aligned}$$

By the above, we also have $\mathrm {RM}_q(d,m):=\{(f(P))_{P \in {\mathbb A}^m} \, : \, f \in T_{\le d}^\mathrm {red}(m)\}$ and moreover, we have

$$\begin{aligned} \dim \mathrm {RM}_q(d,m)=\dim T^\mathrm {red}_{\le d}(m). \end{aligned}$$

(1)

Reed–Muller codes $\mathrm {RM}_q(d,m)$ have been studied extensively for their elegant algebraic properties. Their generalized Hamming weights $d_r(\mathrm {RM}_q(d,m))$ have been determined in [4] by Heijnen and Pellikaan. For a general linear code $C \subseteq \mathbb {F}_q^n$ these are defined as follows:

$$\begin{aligned} d_r(C):=\min _{D \subseteq C: \dim D=r}|\mathrm {supp}(D)|, \end{aligned}$$

where the minimum is taken over all r-dimensional $\mathbb {F}_q$-linear subspaces D of C and where $\mathrm {supp}(D)$ denotes the support of D, that is to say

$$\begin{aligned} \mathrm {supp}(D):=\{i \, : \, \exists \, (c_1,\ldots ,c_n) \in D, \ c_i \not = 0\}. \end{aligned}$$

In case of Reed–Muller codes, there is a direct relation between generalized Hamming weights and the number of common solutions to systems of polynomial equations. Indeed, if $D \subset \mathrm {RM}_q(d,m)$ is spanned by $(f_i(P))_{P \in {\mathbb A}^m}$ for $f_1,\ldots ,f_r \in T_{\le d}^\mathrm {red}(m)$, then $|\mathrm {supp}(D)|=q^m-|\mathsf {Z}(f_1,\ldots ,f_r)|$ where $\mathsf {Z}(f_1, \ldots , f_r):=\{P \in {\mathbb A}^m \,:\, f_1(P)=\cdots =f_r(P)=0\}$ denotes the set of common zeros of $f_1, \ldots , f_r$ in the m-dimensional affine space ${\mathbb A}^m$ over $\mathbb {F}_q$. Therefore, if we define

$$\begin{aligned} \bar{e}^{{\mathbb A}}_r(d,m): = \max \left\{ \left| \mathsf {Z}(f_1, \ldots , f_r) \right| \,:\, f_1, \ldots , f_r \in T^\mathrm {red}_{\le d}(m) \; \text { linearly independent}\right\} , \end{aligned}$$

(2)

then $d_r(\mathrm {RM}_q(d,m))=q^m-\bar{e}^{{\mathbb A}}_r(d,m).$ Note that $T^\mathrm {red}(m)$ is a vector space over $\mathbb {F}_q$ of dimension $q^m$ and that a reduced polynomial has total degree at most $m(q-1)$. Therefore $T^\mathrm {red}(m)=T^\mathrm {red}_{\le m(q-1)}(m)$. This implies in particular that $\mathrm {RM}_q(d,m)=\mathbb {F}_q^{q^m}$ for $d \ge m(q-1)$. Therefore, we will always assume that $d \le m(q-1).$

The result of Heijnen–Pellikaan in [4] on the value of $d_r(\mathrm {RM}_q(d,m))$ can now be restated as follows, see for example [2].

$$\begin{aligned} \bar{e}^{{\mathbb A}}_r(d,m)= \sum _{i=1}^m \mu _i q^{m-i}, \end{aligned}$$

(3)

where $(\mu _1, \ldots , \mu _{m})$ is the r-th m-tuple in descending lexicographic order among all m-tuples $(\beta _1, \ldots , \beta _{m})\in \{0,1,\ldots ,q-1\}^m$ satisfying $\beta _1+ \cdots + \beta _{m} \le d$.

Following the notation in [4], we denote with $\rho _q(d,m)$ the dimension of $\mathrm {RM}_q(d,m)$. Equation (1) implies that $\rho _q(d,m)=\dim (T_{\le d}^\mathrm {red}(m)).$ In particular, we have

$$\begin{aligned} \rho _q(d,m)=\dim (T_{\le d}(m))=\left( {\begin{array}{c}m+d\\ d\end{array}}\right) , \quad \hbox {if}\,\, d \le q-1, \end{aligned}$$

(4)

since $T_{\le d}(m)=T_{\le d}^\mathrm {red}(m)$ if $d<q.$ Here as well as later on we use the convention that $\left( {\begin{array}{c}a\\ b\end{array}}\right) =0$ if $a<b$. In particular we have $\rho _q(d,m)=0$ if $d<0$. As shown in [1, §5.4], for the general case $d \le m(q-1)$, we have

$$\begin{aligned} \rho _q(d,m)=\dim \left( T^\mathrm {red}_{\le d}(m)\right) =\sum _{i=0}^d \sum _{j=0}^{m} (-1)^j \left( {\begin{array}{c}m\\ j\end{array}}\right) \left( {\begin{array}{c}m-1+i-qj\\ m-1\end{array}}\right) . \end{aligned}$$

(5)

In this note, we will present an easy-to-obtain expression for $\bar{e}^{{\mathbb A}}_r(d,m)$ involving a certain representation of the number $\rho _q(d,m)-r$ that we introduce in the next section.

2 The d-th Macaulay representation with respect to q

Let d be a positive integer. The d-th Macaulay (or d-binomial) representation, of a nonnegative integer N is a way to write N as sum as certain binomial coefficients. To be precise

$$\begin{aligned} N=\sum _{i=1}^d \left( {\begin{array}{c}s_i\\ i\end{array}}\right) , \end{aligned}$$

where the $s_i$ integers satisfying $s_d>s_{d-1}> \cdots > s_1 \ge 0$. The usual convention that $\left( {\begin{array}{c}a\\ b\end{array}}\right) =0$ if $a<b$, is used. For example, the d-th Macaulay representation of 0 is given by $0=\sum _{i=1}^d \left( {\begin{array}{c}i-1\\ i\end{array}}\right) .$ Given d and N the integers $s_i$ exist and are unique. The Macaulay representation is among other things used for the study of Hilbert functions of graded modules, see for example [3]. It is well known (see for example [3]) that if N and M are two nonnegative integers with Macaulay representations given by $(k_d,\ldots ,k_1)$ and $(\ell _d,\ldots ,\ell _1)$ then $N \le M$ if and only if $(k_d,\ldots ,k_1) \preccurlyeq (\ell _d,\ldots ,\ell _1)$, where $\preccurlyeq$ denotes the lexicographic order.

For our purposes it is more convenient to define $m_i:=s_i-i$. We then obtain

$$\begin{aligned} N=\sum _{i=1}^d \left( {\begin{array}{c}m_i+i\\ i\end{array}}\right) , \end{aligned}$$

(6)

where $m_i$ are integers satisfying $m_d \ge m_{d-1} \ge \cdots \ge m_1 \ge -1.$ The reason for this is that for $d \le q-1$ we have $\rho _q(d,m)=\left( {\begin{array}{c}m+d\\ d\end{array}}\right)$. Therefore, we can interpret Eq. (6) as a statement concerning dimensions of the Reed–Muller codes $\mathrm {RM}_q(i,m_i)$. For a suitable choice of N, it turns out that the $m_i$ completely determine the value of $\bar{e}^{{\mathbb A}}_r(d,m)$ if $d \le q-1$. For $d \ge q$, even though the dimension $\rho _q(d,m)$ is not longer given by $\left( {\begin{array}{c}m+d\\ d\end{array}}\right)$, there exists a variant of the usual d-th Macaulay representation that turns out to be equally meaningful for Reed–Muller codes. Before stating this representation, we give a lemma.

Lemma 2.1

Let $m \ge 1$ be an integer. We have

$$\begin{aligned} \rho _q(d,m) = \sum _{i=0}^{\min \{d,q-1\}}\rho _q(d-i,m-1). \end{aligned}$$

Proof

Any polynomial $f \in T(m)$ can be seen as a polynomial in the variable $X_m$ with coefficients in $T(m-1)$. This implies that $T(m) = \sum _{i \ge 0} X_m^i T(m)$, where the sum is a direct sum. Similarly we can write

$$\begin{aligned} T^\mathrm {red}_{\le d}(m)=\sum _{i=0}^{\min \{d,q-1\}}X_m^iT^\mathrm {red}_{\le d-i}(m-1). \end{aligned}$$

The result now follows.$\square$

A consequence of this lemma is the following.

Corollary 2.2

Let $d=a(q-1)+b$ for integers a and b satisfying $a \ge 0$ and $1 \le b \le q-1.$ Further suppose that $m \ge a$. Then

$$\begin{aligned} \rho _q(d,m)-1=\sum _{j=0}^{a-1} \sum _{\ell =0}^{q-2} \rho _q(d-j(q-1)-\ell ,m-j-1) + \sum _{i=1}^b\rho _q(i,m-a-1). \end{aligned}$$

Proof

This follows using Lemma 2.1 repeatedly. First applying the lemma to each sum within the double summation on the right-hand side, we see that

$$\begin{aligned}&\sum _{j=0}^{a-1} \sum _{\ell =0}^{q-2} \rho _q(d-j(q-1)-\ell ,m-j-1) \\&\quad =\sum _{j=0}^{a-1} \left( \rho _q(d-j(q-1),m-j)-\rho _q(d-(j+1)(q-1),m-j-1) \right) \\&\quad =\rho _q(d,m)-\rho _q(d-a(q-1),m-a) = \rho _q(d,m)-\rho _q(b,m-a). \end{aligned}$$

Using the same lemma to rewrite the single summation on the right-hand side in Eq. (9) we see that if $m>a$

$$\begin{aligned} \sum _{i=1}^b\rho _q(i,m-a-1)=\rho _q(b,m-a)-\rho _q(0,m-a-1)=\rho _q(b,m-a)-1, \end{aligned}$$

while if $m=a$, the single summation equals 0 and the double summation simplifies to $\rho _q(d,m)-1$. In either case, we obtain the desired result $\square$

We can now show the following.

Theorem 2.3

Let $N \ge 0$ and $d \ge 1$ be integers and q a prime power. Then there exist uniquely determined integers $m_1,\ldots ,m_d$ satisfying

• 1. $N=\sum _{i=1}^d \rho _q(i,m_i),$

• 2. $-1 \le m_1 \le \cdots \le m_d,$

• 3. for all i satisfying $1 \le i \le d-q+1$, either $m_{i+q-1} > m_i$ or $m_{i+q-1}=m_i=-1$.

Proof

We start by showing uniqueness. Suppose that

$$\begin{aligned} N=\sum _{i=1}^d \rho _q(i,m_i)=\sum _{i=1}^d \rho _q(i,n_i) \end{aligned}$$

(7)

and the integers $n_1,\ldots ,n_d$ and $m_1,\ldots m_d$ satisfy the conditions from the theorem. First of all, if $m_d=-1$ or $n_d=-1$ then $N=0$. Either assumption implies that $(m_d,\ldots ,m_1)=(-1,\ldots ,-1)=(n_d,\ldots ,n_1)$. Indeed $n_i\ge 0$ or $m_i \ge 0$ for some i directly implies that $N>0$. Therefore we from now on assume that $m_d\ge 0$ and $n_d \ge 0$. To arrive at a contradiction, we may assume without loss of generality that $n_d \le m_d-1$.

Define e to be the smallest integer such that $n_e \ge 0$. Equation (7) can then be rewritten as

$$\begin{aligned} N=\sum _{i=1}^d \rho _q(i,m_i)=\sum _{i=e}^d \rho _q(i,n_i) \end{aligned}$$

(8)

Condition 3 from the theorem implies that $n_{i-q+1}<n_i$ for all i satisfying $e \le i \le d$. Now write $d-e+1=a(q-1)+b$ for integers a and b satisfying $a \ge 0$ and $1 \le b \le q-1$. With this notation, we obtain that for any $0 \le j \le a-1$ and $0 \le \ell \le q-2$ we have that

$$\begin{aligned} n_{d-j(q-1)-\ell } \le n_d-j \le m_d-j-1. \end{aligned}$$

In particular choosing $j=a-1$ and $\ell =0$, this implies that $m_d \ge a+n_{q-1+b} \ge a+1+n_b \ge a$. Using these observations, we obtain from Eq. (7) that

$$\begin{aligned} \rho _q(d,m_d) \le N= & {} \sum _{i=e}^d \rho _q(i,n_i)\nonumber \\\le & {} \sum _{j=0}^{a-1} \sum _{\ell =0}^{q-2} \rho _q(d-j(q-1)-\ell ,m_d-j-1)\nonumber \\&+\sum _{i=1}^b\rho _q(e+i-1,m_d-a-1). \end{aligned}$$

(9)

Applying the same technique as in the proof of Corollary 2.2, we derive that

$$\begin{aligned} \sum _{j=0}^{a-1} \sum _{\ell =0}^{q-2} \rho _q(d-j(q-1)-\ell ,m_d-j-1)=\rho _q(d,m_d)-\rho _q(b+e-1,m_d-a) \end{aligned}$$

and Eq. (9) can be simplified to

$$\begin{aligned} \rho _q(d,m_d) \le \rho _q(d,m_d)-\rho _q(b+e-1,m_d-a) + \sum _{i=1}^b\rho _q(e+i-1,m_d-a-1). \end{aligned}$$

(10)

For $m_d=a$ the right-hand side equals $\rho _q(d,m_d)-1$, leading to a contradiction. If $m_d>q$, Eq. (10) implies

$$\begin{aligned} \rho _q(b+e-1,m_d-a)\le & {} \sum _{i=1}^b\rho _q(e+i-1,m_d-a-1)\\= & {} \sum _{j=0}^{b-1}\rho _q(e+b-1-j,m_d-a-1)\\< & {} \sum _{j=0}^{\min \{e+b-1,q-1\}}\rho _q(e+b-1-j,m_d-a-1)\\= & {} \rho _q(b+e-1,m_d-a), \end{aligned}$$

where in the last equality we used Lemma 2.1. Again we arrive at a contradiction. This completes the proof of uniqueness of the d-th Macaulay representation with respect to q.

Now we show existence. Let d, N and q be given. We will proceed with induction on d. For $d=1$, note that $\rho _q(1,m)=m+1$ for any $m \ge -1$. Therefore, for a given $N \ge 0$, we can write $N=\rho _q(1,N-1)$.

Now assume the theorem for $d-1$. There exists $m_d \ge -1$ such that

$$\begin{aligned} \rho _q(d,m_d) \le N < \rho _q(d,m_d+1). \end{aligned}$$

(11)

Applying the induction hypothesis on $N-\rho _q(d,m_d)$, we can find $m_{d-1},\ldots ,m_1$ satisfying the conditions of the theorem for $d-1$. In particular we have that

• 1. $N-\rho _q(d,m_d)=\sum _{i=1}^{d-1} \rho _q(i,m_i),$

• 2. $-1 \le m_1 \le \cdots \le m_{d-1},$

• 3. $m_{i+(q-1)} > m_i$ for all $1 \le i \le d-q.$

Clearly this implies that $N=\sum _{i=1}^{d} \rho _q(i,m_i),$ but it is not clear a priori that $m_1,\ldots ,m_d$ satisfy conditions 2 and 3 as well. Conditions 2 and 3 would follow once we show that $m_d \ge m_{d-1}$ and either $m_d > m_{d-q+1}$ or $m_d=m_{d-q+1}=-1$. First of all, if $m_d=-1$, then $N=0$ and $(m_d,\ldots ,m_1)=(-1,\ldots ,-1)$. Hence there is nothing to prove in that case. Assume $m_d \ge 0$. From Eq. (11) and Lemma 2.1 we see that

$$\begin{aligned} N-\rho _q(d,m_d)<\rho _q(d,m_d+1)-\rho _q(d,m_d)= \sum _{i=1}^{\min \{d,q-1\}}\rho _q(d-i,m_d). \end{aligned}$$

(12)

First suppose that $d \le q-1$. First of all, Condition 3 is empty in that setting. Further, Eq. (12) implies

$$\begin{aligned} N-\rho _q(d,m_d) < \sum _{i=1}^{d}\rho _q(d-i,m_d) = \sum _{i=1}^{d-1}\rho _q(d-i,m_d) +1 \end{aligned}$$

and hence

$$\begin{aligned} N-\rho _q(d,m_d) \le \sum _{i=1}^{d-1}\rho _q(d-i,m_d)= \sum _{j=0}^{d-2}\rho _q(d-1-j,m_d) < \rho _q(d-1,m_d+1). \end{aligned}$$

This shows that $m_{d-1} \le m_d$ as desired.

Now suppose that $d \ge q$. In this situation Eq. (12) implies

$$\begin{aligned} N-\rho _q(d,m_d)< \sum _{i=1}^{q-1}\rho _q(d-i,m_d) = \sum _{j=0}^{q-2}\rho _q(d-1-j,m_d)< \rho _q(d-1,m_d+1). \end{aligned}$$

Hence $m_{d-1} \le m_d$ as before. Finally assume that $m_d \le m_{d-q+1}$. Then by the previous and Condition 2, we have $m_{d}= m_{d-1}=\cdots = m_{d-q+1}$. Hence $N \ge \sum _{i=0}^{q-1}\rho _q(d-i,m_d)=\rho _q(d,m_d+1)$ which is in contradiction with Eq. (11). This concludes the induction step and hence the proof of existence. $\square$

We call the representation of N in the above theorem the d-th Macaulay representation of N with respect to q. One retrieves the usual d-th Macaulay representation letting q tend to infinity. We refer to $(m_d,\ldots ,m_1)$ as the coefficient tuple of this representation. A direct corollary of the above is the following.

Corollary 2.4

The coefficient tuple $(m_d,\ldots ,m_1)$ of the d-th Macaulay representation with respect to q of a nonnegative integer N can be computed using the following greedy algorithm: The coefficient $m_{d-i}$ can be computed recursively (starting with $i=0$) as the unique integer $m_{d-i} \ge -1$ such that

$$\begin{aligned} \rho _q(d-i,m_{d-i}) \le N-\sum _{j=d-i+1}^{d}\rho _q(j,m_j) <\rho _q(d-i,m_{d-i}+1). \end{aligned}$$

Proof

From the existence-part of the proof of Theorem 2.3 it follows directly that the given greedy algorithm finds the desired coefficients. $\square$

A further corollary is the following. As before $\preceq$ denotes the lexicographic order.

Corollary 2.5

Suppose the N and M are two nonnegative integers whose respective coefficient tuples are $(n_d,\ldots ,n_1)$ and $(m_d,\ldots ,m_1)$. Then

$$\begin{aligned} N \le M \quad \hbox {if and only if} \,\, (n_d,\ldots ,n_1) \preceq (m_d,\ldots ,m_1). \end{aligned}$$

Proof

Assume $(n_d,\ldots ,n_1) \preceq (m_d,\ldots ,m_1).$ It is enough to show the corollary in case $n_d < m_d$. We know from the previous corollary that $n_d$ and $m_d$ may be determined using the given greedy algorithm. In particular this implies that $n_d < m_d$ implies

$$\begin{aligned} N < \rho _q(d,n_d+1) \le \rho _q(d,m_d) \le M. \end{aligned}$$

Assume that $N \le M$. We use induction on d. The induction basis is trivial: If $d=1$, then $m_1=M-1$ and $n_1=N-1$. For the induction step, note that $N \le M < \rho _q(d,m_d+1)$ implies by the greedy algorithm that $n_d \le m_d$. If $n_d < m_d$, we are done. If $n_d=m_d$, we replace N with $N-\rho _q(d,m_d)$ and M with $M-\rho _q(d,m_d)$ and use the induction hypothesis to conclude that $(n_d,\ldots ,n_1) \preceq (m_d,\ldots ,m_1)$. $\square$

3 A simple expression for $\bar{e}^{{\mathbb A}}_r(d,m)$

We are now ready to state and prove the relation between the Macaulay representation with respect to q and $\bar{e}^{{\mathbb A}}_r(d,m)$.

Theorem 3.1

For $1 \le r \le \rho _q(d,m)$, let the d-th Macaulay representation of $\rho _q(d,m)-r$ with respect to q be given by

$$\begin{aligned} \rho _q(d,m)-r=\sum _{i=1}^d \rho _q(i,m_i). \end{aligned}$$

Denoting the floor function as $\lfloor \cdot \rfloor$, we have

$$\begin{aligned} \bar{e}^{{\mathbb A}}_r(d,m)=\sum _{i=1}^d \lfloor q^{m_i} \rfloor . \end{aligned}$$

Proof

We know from Eq. (3) that we need to show that

$$\begin{aligned} \sum _{i=1}^d \lfloor q^{m_i} \rfloor =\sum _{i=1}^m \mu _i q^{m-i}, \end{aligned}$$

with $(\mu _1, \ldots , \mu _{m})$ is the r-th element in descending lexicographic order among all m-tuples $(\beta _1,\ldots ,\beta _m)$ in $\{0,1,\ldots ,q-1\}^m$ satisfying $\beta _1+ \cdots + \beta _{m} \le d$. First of all note that since $r \ge 1$, we have $\rho _q(d,m)-r < \rho _q(d,m)$. In particular this implies that $m_d \le m-1$. Therefore the coefficients of the d-tuple $(m_d,\ldots ,m_1)$ are in $\{-1,0,\ldots ,m-1\}$. Now for $1 \le i \le m+1$ define $\mu _i:=|\{j \, : \, m_j=m-i\}|.$ Since the d-tuple $(m_d,\ldots ,m_1)$ is nonincreasing by Condition 2 from Theorem 2.3, we can reconstruct it uniquely from the $(m+1)$-tuple $(\mu _{1},\mu _2,\ldots ,\mu _{m+1}).$ Moreover, Condition 3 from Theorem2.3, implies that $(\mu _1,\ldots ,\mu _m) \in \{0,1,\ldots ,q-1\}^m$, but note that $\mu _{m+1}$ could be strictly larger than $q-1$. Further by construction we have $\mu _1+\cdots +\mu _m+\mu _{m+1}=d$, implying that $\mu _1+\cdots +\mu _m \le d$. Note that $\mu _{m+1}$ is determined uniquely by $(\mu _1,\ldots ,\mu _m)$, since $\mu _0=d-\mu _1-\cdots -\mu _m$. Therefore the correspondence between the d-tuples $(m_d,\ldots ,m_1)$ of coefficients of the d-th Macaulay representations with respect to q of integers $0 \le N < \rho _q(d,m)$ and the m-tuples $(\mu _1, \ldots , \mu _{m})\in \{0,1,\ldots ,q-1\}^m$ satisfying $\mu _1+ \cdots + \mu _{m} \le d$, is a bijection. Moreover by construction we have

$$\begin{aligned} \sum _{i=1}^d\lfloor q^{m_i} \rfloor =\sum _{j=1}^{m+1} \mu _j \lfloor q^{m-j} \rfloor = \sum _{j=1}^{m} \mu _j q^{m-j}. \end{aligned}$$

What remains to be shown is that the constructed m-tuple coming from the integer $\rho _q(d,m)-r$ is in fact the r-th in descending lexicographic order. First of all, by Corollary 2.2 we see that for $r=1$ and $d=aq+b$ that the m-tuple associated to $\rho _q(d,m)-1$ equals $(q-1,\ldots ,q-1,b,0,\ldots ,0)$, which under the lexicographic order is the maximal m-tuple among all m-tuples $(\beta _1, \ldots , \beta _{m})\in \{0,1,\ldots ,q-1\}^m$ satisfying $\beta _1+ \cdots + \beta _{m} \le d$. Next we show that the conversion between d-tuples $(m_d,\ldots ,m_1)$ to m-tuples $(\mu _1,\ldots ,\mu _m)$ preserves the lexicographic order. Suppose therefore that $1\le r \le s \le \rho _q(d,m)$. We write $N:=\rho _q(d,m)-s$ and $M:=\rho _q(d,m)-r.$ and denote their Macaulay coefficient tuples with $(n_d,\ldots ,n_1)$ and $(m_d,\ldots ,m_1)$. Since $N \le M$, Corollary 2.5 implies that $(n_d,\ldots ,n_1) \preceq (m_d,\ldots ,m_1)$. Also, since these d-tuples are nonincreasing, this implies that their associated m-tuples $(\nu _1,\ldots ,\nu _m)$ and $(\mu _1,\ldots ,\mu _m)$ satisfy $(\nu _1,\ldots ,\nu _m) \preceq (\mu _1,\ldots ,\mu _m)$. Indeed assuming without loss of generality that $\nu _1<\mu _1$ we see that $m_i=n_i=m-1$ for $d-\nu _1 \le i \le d$ but $n_i < m_i=m-1$ for $i=\nu _1+1$. Now the desired result follows immediately. $\square$

Combining this theorem with the greedy algorithm in Corollary 2.4, it is very simple to compute values of $\bar{e}^{{\mathbb A}}_r(d,m)$ or equivalently of $d_r(\mathrm {RM}_q(d,m))$. We illustrate this in the two following examples. The parameters in these example also occur in examples from [4].

Example 3.2

Let $q=4$, $r=8$, $d=m=3$. Since $d\le q-1$, we may work with the usual Macaulay representation when applying Theorem 3.1. We have $\rho _q(d,m)=\left( {\begin{array}{c}6\\ 3\end{array}}\right) =20$ and hence

$$\begin{aligned} \rho _q(d,m)-r=12=\left( {\begin{array}{c}5\\ 3\end{array}}\right) +\left( {\begin{array}{c}2\\ 2\end{array}}\right) +\left( {\begin{array}{c}1\\ 1\end{array}}\right) = \rho _4(3,2)+\rho _4(2,0)+\rho _4(1,0) \end{aligned}$$

is the 3-rd Macaulay representation of 12. Theorem 3.1 implies that $\bar{e}^{{\mathbb A}}_8(3,3)=4^2+4^0+4^0=18$ and hence $d_8(\mathrm {RM}_4(3,3))=64-18=46$ in accordance with Example 6.10 in [4].

Example 3.3

Let $q=2$, $r=10$, $d=3$ and $m=5$. We have $\rho _2(3,5)=26$ by Eq. (5) and hence applying the greedy algorithm from Corollary 2.4, we compute that

$$\begin{aligned} \rho _q(d,m)-r=16=15+1+0=\rho _2(3,4)+\rho _2(2,0)+\rho _2(1,-1) \end{aligned}$$

is the 3rd Macaulay representation of 16 with respect to 2. Theorem 3.1 implies that $\bar{e}^{{\mathbb A}}_{10}(3,3)=2^4+2^0=17$ and hence $d_8(\mathrm {RM}_2(3,5))=32-17=15$ in accordance with Example 6.12 in [4].

Remark 3.4

Theorem 3.1 is somewhat similar in spirit as Theorem 6.8 from [4] in the sense that in both theorems a certain representation in terms of dimensions of Reed–Muller codes is used to give an expression for $d_r(\mathrm {RM}_q(d,m))$. Where we studied decompositions of $\rho _q(d,m)-r$, in [4] the focus was on r itself. This suggest there may exist a duality between the two approaches, but the similarities seem to stop there. The representation in [4] is not the Macaulay representation with respect to q that we have used here. For us it is for example very important that each degree i between 1 and d occurs once in Theorem 2.3 (implying that the greedy algorithm terminates after at most d iterations), while this is not the case in Theorem 6.8 [4]. It could be interesting future work to determine if a deeper lying relationship between the two approaches exists.