1 Introduction

We assume familiarity with the basic facts and notions from error-correcting codes and combinatorial designs and Hadamard matrices [1, 4, 9, 12]. All codes in this paper are ternary. A monomial matrix with entries from GF(3) is a square matrix such that every row and every column contains exactly one nonzero entry. By an automorphism group of a ternary code we mean monomial automorphism group, unless specified otherwise. The permutation automorphism group of a code is the subgroup of its monomial automorphism group that consists of coordinate permutations only.

A Hadamard matrix of order n is an \(n \times n\) matrix H of 1’s and \(-1\)’s such that \(HH^T=nI\), where I is the identity matrix. It follows that \(n=1, 2\), or \(n=4t\) for some integer \(t\ge 1\). An automorphism of a Hadamard matrix H is a pair of \(\{ 0,1,-1\}\)-monomial matrices L, R such that \(LHR=H\). Two Hadamard matrices \(H_1\), \(H_2\) of the same order are equivalent if there are monomial matrices L, R such that \(LH_{1}R=H_2\). A Hadamard matrix H is normalized with respect to its ith row and jth column if all entries in row i and column j are equal to 1. If H is a Hadamard matrix of order \(n=4t\) that is normalized with respect to row i and column j, deleting the ith row and the jth column and replacing all \(-1\)’s with zeros gives the (0, 1)-incidence matrix of a symmetric 2-\((4t-1,2t-1,t-1)\) design D called a Hadamard 2-design, while deleting the jth column of H and the jth column of \(-H\) from the matrix \((H,-H)\) gives the point-by-block \((\pm 1)\)-incidence matrix of a 3-\((4t,2t,t-1)\) design \(D^*\), called a Hadamard 3-design obtained from H with respect to column j. The design D is the derived design of \(D^*\) with respect to its ith point. A Hadamard matrix H of order \(n=4t\) is regular if all rows of H contain the same number k of \(-1\)’s. It follows that \(t=m^2\) for some integer m, \(k=2m^2 \pm m\), and replacing all \(-1\)’s with zeros gives the (0, 1)-incidence matrix of a symmetric 2-\((4m^2,2m^2\pm t,m^2\pm m)\) design. For more on Hadamard matrices and related designs, see, for example, [1, Chapter 7], [9, Chapter 14], [12, Sect. 8.9].

Let q be an odd prime power such that \(q \equiv -1 \pmod 3\). The Pless symmetry code C(q) [22, 23] of length \(n=2q+2\) is a ternary self-dual code with a generator matrix

$$\begin{aligned} G=(I_{q+1}, S_q), \end{aligned}$$
(1)

where \(I_{q+1}\) is the identity matrix of order \(q+1\), and \(S_q =(s_{i,j})\) is a \((q+1)\times (q+1)\) matrix defined as follows. The rows and columns of \(S_q\) are labeled by \(\infty \) and the q elements of the finite field GF(q) of order q, where \(s_{\infty ,\infty }=0\), \(s_{a,a}=0\), \(s_{\infty ,a}=1\) for \(a\in GF(q)\), \(s_{a,\infty }=1\) if \(-1\) is a square in GF(q), and \(s_{a,\infty }=-1\) if \(-1\) is not a square in GF(q) for \(a\in GF(q)\), and \(s_{a,b} =1\) for \(a, b\in GF(q)\) such that \(a\ne b\) and \(a-b\) is a square in GF(q), and \(s_{a,b} =-1\) for \(a, b\in GF(q)\) such that \(a\ne b\) and \(a-b\) is not a square in GF(q). For example, if \(q=5\), the rows and columns of \(S_5\) are labeled by \(\infty , 0, 1, 2, 3, 4\), and

$$\begin{aligned} S_5 =\left( \begin{array}{rrrrrr} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 &{} -1 &{} - 1 &{} 1\\ 1 &{} 1 &{} 0 &{} 1 &{} -1 &{} -1\\ 1 &{} -1 &{} 1 &{} 0 &{} 1 &{} -1\\ 1 &{} -1 &{} -1 &{} 1 &{} 0 &{} 1\\ 1 &{} 1 &{} -1 &{} -1 &{} 1 &{} 0 \end{array} \right) . \end{aligned}$$

The main property of \(S_q\) is that \(S_{q}S_{q}^T=qI_{q+1}\), which implies \(S_{q}S_{q}^T \equiv -I_{q+1} \pmod 3\); hence C(q) is self-dual. The symmetry codes C(5)Footnote 1 , C(11), C(17), C(23) and C(29) are extremal self-dual codes of length n divisible by 12 and minimum distance d meeting the Mallows-Sloane upper bound \(d\le 3[n/12]+3\) [16]; hence these codes support 5-designs by the Assmus–Mattson Theorem [3].

Pless [23] proved that in addition to the trivial monomial automorphism of order 2 corresponding to the negation of all code coordinates, the monomial automorphism group of C(q) contains a subgroup of order \(q(q^2 -1)\) isomorphic to \(PGL_{2}(q)\). In addition, it was proved in [23, Theorem 4.2] that the symmetry code C(q) contains a set of \(2q+2\) codewords of weight \(2q+2\) that form a Hadamard matrix of order \(2q+2\). Specifically, if \(q\equiv 1 \pmod 4\) the Hadamard matrix formed by codewords of weight \(2q+2\) is

$$\begin{aligned} H_{1}(q) =\left( \begin{array}{rr} I + S_q &{} -I +S_q\\ -I + S_q &{} -I - S_q \end{array}\right) , \end{aligned}$$
(2)

while if \(q\equiv -3 \pmod 4\) the Hadamard matrix is

$$\begin{aligned} H_3(q) = \left( \begin{array}{rr} I + S_q &{} I +S_q\\ I - S_q &{} -I + S_q \end{array}\right) , \end{aligned}$$
(3)

where I is the identity matrix of order \(q+1\).

The Hadamard matrices (2), (3) are known in the combinatorial literature as Paley–Hadamard matrices of type II [7, 9, 21, 14.1]. If \(q\equiv 3 \pmod 4\), the \((q+1)\times (q+1)\) matrix obtained by bordering the matrix \(S_q -I\) with one all-one row and one all-one column is a Hadamard matrix of order \(q+1\), known as a Paley–Hadamard matrix, or a Paley–Hadamard matrix of type I [9, 14.1], [7, 21]. The unique (up to equivalence) Hadamard matrix of order 12 is both a Paley–Hadamard matrix of type I for \(q=11\) and a Paley–Hadamard matrix of type II for \(q=5\), and its full automorphism group modulo its center of order 2 is the Mathieu group \(M_{12}\) (Hall [8]). The full automorphism group of a Paley–Hadamard matrix of type I for \(q>11\) was determined by Kantor [13] and is of order \(q(q^2-1)\), while the full automorphism group of a Paley–Hadamard matrix of type II for \(q>5\) was determined by de Launey and Stafford [7], and is of order \(4fq(q^2 -1)\) if \(q=p^f\), where p is prime.

If \(q=5\), 11, or 23, the number of all codewords of full weight \(2q+2\) in the symmetry code C(q) is exactly \(4q+4\) [23]. These codewords span the code and consist of the rows of the Hadamard matrix \(H_1(q)\) from (2) (resp. \(H_3(q)\) from (3)) and its negative, or \(2H_1(q)\) (resp. \(2H_2(q)\)); hence the full monomial automorphism group of C(q) coincides with the full automorphism group of \(H_1(q)\) (resp. \(H_3(q)\)) [23].

The symmetry code C(17) of length 36 contains 888 codewords of weight 36; hence it is not clear whether the automorphism group of the Hadamard matrix \(H_1(17)\) (2) is the full automorphism group of C(17). In Sect. 2, we prove that the full automorphism group of C(17) coincides with the full automorphism group of \(H_1(17)\), the latter being a Paley–Hadamard matrix of type II; hence its order is \(4\cdot {17}(17^2 -1)=19584\). In addition, we classify all Hadamard matrices of order 36 having as rows codewords of C(17) of weight 36, and show that up to equivalence, there are exactly two such matrices: \(H_1(17)\), and a second Hadamard matrix \(H'\) having the property that all Hadamard 3-(36, 18, 8) designs associated with \(H'\) are isomorphic and have trivial full automorphism group of order 1. The full automorphism group of \(H'\) is of order 72 and is transitive on the set of 72 rows (as well as the set of 72 columns) of \(H'\) and \(-H'\). The 3-rank of \(H'\) is 18; thus C(17) is the row space of \(H'\). The Hadamard matrix \(H'\) is regular, and the symmetric 2-(36, 15, 6) design with \(\pm \,1\)-incidence matrix \(H'\) has trivial full automorphism group.

In Sect. 3 we discuss Paley–Hadamard matrices of Type I and Hadamard 3-designs arising from extended ternary quadratic residue codes.

2 Hadamard matrices and designs arising from symmetry codes

The sum (over GF(3)) of all rows of the generator matrix (1) of the symmetry code C(q) is a vector v of full Hamming weight \(2q+2\), with all components equal to 1 if \(-1\) is not a square in GF(q), (that is, v is the constant all-one vector \(\bar{1}=(1,\ldots ,1)\)), and v has \(2q+1\) components equal to 1, and the position labeled by \(\infty \) is equal to \(-1\) whenever \(-1\) is a square in GF(q).

Next, we consider a code which is monomially equivalent to the Pless symmetry code C(q), and always contains the all-one vector, namely the code L(q) with a generator matrix \(G'\) given by

$$\begin{aligned} G' =(I_{q+1}, U_q), \end{aligned}$$
(4)

where \(U_q\) is a \((q+1)\times (q+1)\) matrix obtained from \(S_q\) by replacing every nonzero entry in the column labeled by \(\infty \) with \(-1\). Clearly, the generator matrix \(G'\) (4) is identical with the generator matrix G (1) if \(-1\) is not a square in GF(q), and is obtained by negating one column of G if \(-1\) is a square in GF(q). Thus, the matrices (1) and (4) generate monomially equivalent ternary codes.

Using (4), we obtain a parity check P matrix of L(q) given by

$$\begin{aligned} P = (-U_q^T, I_{q+1}). \end{aligned}$$
(5)

Note that since L(q) is self-dual, the rows of P are codewords of L(q). It is easy to check that the matrix H given by

$$\begin{aligned} H=\left( \begin{array}{c} G'+P\\ G' -P \end{array}\right) = \left( \begin{array}{cc} I_{q+1}-U_q^T &{} U_q +I_{q+1}\\ I_{q+1}+U_q^T &{} U_q -I_{q+1} \end{array}\right) \end{aligned}$$
(6)

is a Hadamard matrix of order \(2q+2\), with rows being codewords of L(q).

Theorem 2.1

The code L(q) contains a set of \(4q+2\) (0,1)-codewords of weight \(q+1\) that form the block-by-point incidence matrix of a Hadamard 3-\((2q+2,q+1,(q-1)/2)\) design D(q) associated with a Paley–Hadamard matrix of type II.

Proof

All entries in the first row of the Hadamard matrix H (6) are equal to 1; that is, H is normalized with respect to its first row, and consequently, all entries in the first row of \(-H\) are equal to \(-1\). Adding the constant codeword \(\bar{2}=(2,\ldots ,2)\) with all entries equal to 2 to every row of the matrix

$$\begin{aligned} \left( \begin{array}{rr} H\\ -H \end{array}\right) \end{aligned}$$

gives a (0, 1)-matrix M with all-zero first row, and all-one row labeled by the first row of \(-H\). Deleting the all-zero row and the all-one row from M gives a \((4q+2)\times (2q+2)\) (0, 1)-matrix A, being the block-by-point incidence matrix of a Hadamard 3-\((2q+2,q+1,(q-1)/2)\) design associated with the first row of H. Clearly, H is equivalent to the corresponding matrix (2) or (3); hence H is equivalent to a Paley–Hadamard matrix of type II. \(\square \)

Theorem 2.2

If \(q=5, 11, 17, 23\), the code L(q) contains exactly \(4q+2\) (0,1)-codewords of weight \(q+1\), and every such codeword is the incidence vector of a block of the Hadamard 3-design D(q) from Theorem 2.1.

Proof

Let m denote the total number of (0,1)-codewords of weight \(q+1\) in L(q). By Theorem 2.1, \(m\ge 4q+2\). If \(v\in L(q)\) is a (0,1)-codeword of weight \(q+1\) then \(v+\bar{1}\) is a codeword of full weight \(2q+2\), having \(q+1\) components equal to 1, and \(q+1\) components equal to 2. Adding the codewords \(\bar{1}\) and \(\bar{2} = 2\cdot \bar{1}\) gives \(m+2 \ge 4q+4\) codewords of weight \(2q+2\). Since C(q) contains exactly \(4q+4\) codewords of weight \(2q+2\) for \(q=5, 11, 23\) [23], the statement is true in these cases.

The case \(q=17\) needs additional analysis because the symmetry code C(17) , as well as its equivalent code L(17), contains 888 codewords of weight 36 [23]. The set of all codewords of weight 36 is easily computed with Magma [5]. This set comprises of the following codewords:

  • the 36 rows of the Hadamard matrix H (6), one of the rows being \(\bar{1}\), and 35 rows with 18 components equal to 1, and 18 components equal to \(-1\) (note that \(-1\equiv 2 \pmod 3\));

  • the 36 rows of 2H that include \(\bar{2}\) and 35 rows with 18 components equal to 1, and 18 components equal to 2;

  • a set T of 408 codewords having 15 components equal to 1 and 21 components equal to 2;

  • a set 2T of 408 codewords obtained by multiplying every codeword from T by 2.

Note that adding \(\bar{2}\) to any (0, 1)-codeword of weight 18 gives a codeword of weight 36 with 18 1’s and 18 2’s; hence the code L(17) contains exactly 70 (0, 1)-codewords of weight 18 obtained by adding the codeword \(\bar{2}\) to the rows of H and 2H, and these 70 (0, 1)-codewords form the incidence matrix of the 3-design D(17) from Theorem 2.1. \(\square \)

Note 1

The code L(29) contains 19606 (0,1)-codewords of weight 30. It is an open question whether this set contains the incidence matrices of any Hadamard 3-(60, 30, 14) designs that are not isomorphic to D(29). The number of codewords of weight 60 in L(29) is 41184. It seems likely that there may be Hadamard matrices of order 60 formed by codewords of weight 60 that are not equivalent to the Paley–Hadamard matrix of type II.

Corollary 2.3

If \(q=5, 11, 17, 23\), the full permutation automorphism group of L(q) coincides with the full automorphism group of the Hadamard 3-design D(q) from Theorem 2.1.

Proof

The results of De Launey and Stafford [7] and Norman [20] imply that the full automorphism group of D(q) has order \(q(q-1)\) if \(q>5\) is prime. The full automorphism group of D(5) has order 7920.

Any derived design with respect to a point of a Hadamard 3-\((2q+2,q+1,(q-1)/2)\) design D is a symmetric Hadamard 2-\((2q+1,q,(q-1)/2)\) design \(D'\) of order \(q - (q-1)/2 =(q+1)/2\). Since \(q\equiv -1 \pmod 3\), 3 divides \((q+1)/2\). If 9 does not divide the order \((q+1)/2\) (which is true if \(q=5\), 11, or 23), the rank of the incidence matrix if \(D'\) over GF(3) (or the 3-rank of \(D'\)) is equal to \(q+1\) (see, for example, Assmus and Key [1, 2]), hence the 3-rank of D is \(q+1\) and the code L(q) is spanned by the incidence matrix of D. If \(q=17\), a direct computation shows that the 3-rank of D(17) is 18, hence D(17) spans the code L(17). \(\square \)

Theorem 2.4

(i) The code L(17) contains two equivalence classes of Hadamard matrices of order 36 having as rows codewords of weight 36, with representatives the Hadamard matrix H (6), which is equivalent to a Paley–Hadamard matrix of type II and has full automorphism group of order 19584, and a second Hadamard matrix \(H'\), being a regular Hadamard matrix such that the symmetric 2-(36, 15, 6) design \(D'\) with (0, 1)-incidence matrix \((H' +J)/2\), where J is the \(36\times 36\) all-one matrix, has a trivial automorphism group.

(ii) The row span of the incidence matrix of the 2-(36, 15, 6) design \(D'\) is an extremal ternary [36, 18, 12] code equivalent to the symmetry code C(17).

(iii) The full automorphism group of the code L(17) coincides with the full automorphism group H.

Proof

(i) In the context of Hadamard matrices, we consider the element 2 of GF(3) as \(-1\). Using the notation from the proof of Theorem 2.2, we define a graph \(\Gamma \) having as vertices the 408 codewords from T, where two codewords \(u, v \in T\) are adjacent in \(\Gamma \) if and only if the Hamming distance between u and v is 18, or equivalently, the intersection of the supports of the (0,1)-vectors \(\bar{2}-u\) and \(\bar{2}-v\) is of size 6. Replacing all entries equal to 2 by zero in every vector from T gives a set T(0, 1) of (0,1)-vectors of weight 15. Using the restricted Johnson bound, it is easy to verify that the maximum number codewords in a binary constant weight code of length 36 with codewords of weight 15 and minimum distance 18, is 36. Every set K of 36 vectors from T(0, 1) that meets the Johnson bound corresponds to a clique of size 36 in \(\Gamma \), and the \(36\times 36\) matrix having as rows the vectors from K is the incidence matrix N of a symmetric 2-(36, 15, 6) design (see [30, Theorem 2.4.12, p. 99] or [31, Sect. 3]). Replacing all zeros in N with \(-1\)’s gives a regular Hadamard matrix of order 36. Using the clique finding algorithm Cliquer [19], a quick computer search shows that the graph \(\Gamma \) contains exactly 272 cliques of size 36, or in other words, there are 272 collections of 36 codewords from T that form a Hadamard matrix of order 36. Further analysis with Magma shows that all 272 Hadamard matrices are equivalent to a matrix \(H'\) with a full monomial automorphism group of order 72 that acts transitively on the set of size 72 being the union of the rows of \(H'\) and the rows of \(-H'\).

The incidence matrix of the symmetric 2-(36, 15, 6) design \(D'\) obtained by replacing all \(-1\)-entries of \(H'\) with zeros is listed in the Appendix. The design \(D'\) has a trivial full automorphism group of order 1.

(ii) The 3-rank of the incidence matrix of \(D'\) is 18, and its row span over GF(3) is a ternary [36, 18, 12] code equivalent to the Pless symmetry code.

Parts (iii) was verified by computer using Magma. The full automorphism group of L(17) partitions the set of the 888 codewords of weight 36 into two orbits, of length 72 and 816 respectively, the orbit of length 72 comprised of the rows of H (6) and \(-H\). Thus, the full automorphism group of the code L(17) coincides with the full automorphism group of H, and is of orderFootnote 2\(19584 =2^7 \cdot 3^2 \cdot 17.\) \(\square \)

Note 2

Up to equivalence, there are exactly 11 Hadamard matrices of order 36 with automorphism groups of order divisible by 17 (Tonchev [28]). Each of these matrices spans a ternary self-dual code of length 36, but only the symmetry code C(17) spanned the Paley–Hadamard matrix of type II is extremal, that is, has minimum distance 12, and supports 5-designs. A stronger characterization of the Pless symmetry code C(17) was proved by Huffman [11], namely that up to equivalence, C(17) is the only extremal ternary self-dual code of length 36 that admits a monomial automorphism of order 17.

Note 3

Hadamard matrices and designs are used for the construction of self-orthogonal and self-dual codes over other finite fields. A classical example is the extended binary Golay code generated by a bordered incidence matrix of a symmetric Hadamard 2-(23, 11, 5) design associated with a Paley–Hadamard matrix of type I. Hadamard matrices of order 28 with an automorphism of order 7 [27] were used by Pless and Tonchev [24] for the classification of self-orthogonal codes over GF(7). The Paley–Hadamard matrix of type II of order 28 is the only Hadamard matrix of this order that admits an automorphism of order 13 and yields an extremal binary self-dual code of length 56 [26, 29]. More extremal binary self-dual codes derived from Hadamard matrices of order 28 were found in [6].

3 Hadamard matrices and designs arising from ternary QR codes

The symmetry codes C(11), C(23), and C(29) have siblings with the same parameters and weight distribution, being ternary extended quadratic-residue codes that support 5-designs by the Assmus-Mattson theorem. If \(q\equiv 3 \pmod 4\) is a prime power, a quadratic residue (QR) code of length q is a code spanned by the (0,1)-incidence matrix A of a symmetric Hadamard 2-\((q, (q-1)/2, (q-3)/4)\) design obtained from the Paley–Hadamard matrix of type I, and its extended code is spanned by a matrix obtained by adding one all-one column to A. If, in addition, \(q\equiv -1 \pmod 3\), that is, q is of the form \(q=12s+11\) for some integer \(s\ge 0\), the ternary extended QR code is self-dual.

Theorem 3.1

Let \(q=12s+11\) be a prime power, and let \(QR_q\) be the ternary extended QR code of length \(q+1\).

(i) \(QR_q\) contains a Paley–Hadamard matrix of type I having as rows codewords of weight \(q+1\).

(ii) \(QR_q\) contains a set of 2q (0,1)-codewords of weight \((q+1)/2\) that form the incidence matrix of a Hadamard 3-\((q+1,(q+1)/2, (q-3)/4)\) design associated with the Paley–Hadamard matrix of type I of order \(q+1\).

(iii) If \(q=11\), 23 or 47, \(QR_q\) contains exactly 2q (0,1)-codewords of weight \((q+1)/2\), and the permutation automorphism group of the code coincides with the full automorphism group of the Hadamard 3-\((q+1,(q+1)/2, (q-3)/4)\) design from part (ii).

Proof

(i) The statement (i) is implicit in [3]. The column sum of the (0,1)-incidence A of the Hadamard 2-\((q, (q-1)/2, (q-3)/4)\) design obtained from the Paley–Hadamard matrix of type I is

$$\begin{aligned} (q-1)/2 =(12s+10)/2 \equiv -1 \pmod 3, \end{aligned}$$

and the sum of all q entries of the all-one column is \(q\equiv -1 \pmod 3\), hence the sum over GF(3) of all rows of the \(q\times (q+1)\)-matrix B obtained by bordering A with one all-one column, is equal to the constant vector \(\bar{2}\). Since \(QR_q\) is the row span of B, the constant vectors \(\bar{1}\) and \(\bar{2}\) belong to the code. Let E be the \((q+1)\times (q+1)\) matrix obtained from B by adding one extra all-one row. The matrix \(H_{q+1}=2J-E\), where J is the \((q+1)\times (q+1)\) all-one matrix, is a Paley–Hadamard matrix of type I. Every row of \(H_{q+1}\) is the difference of the codeword \(\bar{2}\) and a row of B, hence the rows of \(H_{q+1}\) belong to the code \(QR_q\).

(ii) Adding the codeword \(\bar{2}\) to every row of

$$\begin{aligned} \left( \begin{array}{rr} H_{q+1}\\ -H_{q+1} \end{array}\right) \end{aligned}$$
(7)

gives a \((2q+2)\times (q+1)\) matrix with one all-zero row, one all-one row, and 2q (0,1)-rows of weight \((q+1)/2\) that form the incidence matrix of a Hadamard 3-\((q+1,(q+1)/2, (q-3)/4)\) design associated with \(H_{q+1}\).

(iii) The proof is similar to that of Corollary 2.3. \(\square \)

Note 4

The number of codewords of weight 60 in \(QR_{59}\) is 41184. It seems likely that there may be Hadamard matrices of order 60 formed by codewords of weight 60 that are not equivalent to the Paley–Hadamard matrux of type I from Theorem 3.1.

4 Concluding remarks

The extended ternary Golay code of length 12, the Pless symmetry codes C(q) (\(q=11, 17, 23\) and 29), of lengths 24, 36, 48 and 60, the extended ternary QR codes of lengths 24, 48 and 60, and an extremal code of length 60 discovered by Nebe and Villard [18] as an analogue of the Pless symmetry code C(29), are the only known extremal ternary self-dual codes of length divisible by 12 that support 5-designs. It is known that the symmetry code of length 84 (\(q=41\)), as well as the extended QR code of this length are not extremal. Extremal ternary self-dual codes of length n divisible by 12 do not exist for \(n=72, 96, 120\), and all \(n\ge 144\), because then the extremal Hamming weight enumerator contains a negative coefficient [25].

All ternary self-dual codes of length 24 have been classified up to equivalence (Harada and Munemasa [10]), and the symmetry code C(11) and the extended QR code are the only extremal codes of this length. Nine of the self-dual ternary codes of length 24 are spanned by Hadamard matrices of order 24 [14, 15], but only two codes, \(QR_{23}\) and C(11), that are spanned by the Paley–Hadamard matrices of type I and II respectively, are extremal.

It is an interesting open question whether the Pless symmetry codes of length 36, 48, and 60, the extended QR codes of lengths 48 and 60, and the extremal code of length 60 found by Nebe and Villard [18] are the only extremal self-dual codes of these lengths. The results from Sect. 2 show that the symmetry code of length 36 can be obtained from a Hadamard matrix that is not a Paley–Hadamard matrix of type II, and a natural question that arises is whether any other extremal codes of length 36, 48, or 60 can be obtained form Hadamard matrices that are not of Paley type.

The extremal ternary self-dual codes of lengths \(n\ge 36\) have not been classified up to equivalence. A partial classification of such codes of length \(n\le 40\) admitting automorphisms of prime order \(p\ge 5\) was given by Huffman [11]. In addition, it was proved by Nebe [17] that, up to equivalence, the only extremal ternary self-dual codes of length 48 that admit an automorphism of a prime order \(p\ge 5\), are the Pless symmetry code and the extended QR code.