Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices

Finite Fields and Their Applications 55 (2019) 64–77

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices Dean Crnković, Ronan Egan, Andrea Švob ∗ Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia

a r t i c l e

i n f o

Article history: Received 6 March 2018 Received in revised form 4 September 2018 Accepted 7 September 2018 Available online xxxx Communicated by Dieter Jungnickel MSC: 05B20 05B30 94B05 12E20 Keywords: Weighing matrix Orbit matrix Hermitian self-orthogonal code

a b s t r a c t We define the notion of an orbit matrix with respect to standard weighing matrices, and with respect to types of weighing matrices with entries in a finite field. In the latter case we primarily restrict our attention the fields of order 2, 3 and 4. We construct self-orthogonal and Hermitian self-orthogonal linear codes over finite fields from these types of weighing matrices and their orbit matrices respectively. We demonstrate that this approach applies to several combinatorial structures such as Hadamard matrices and balanced generalized weighing matrices. As a case study we construct self-orthogonal codes from some weighing matrices belonging to some well known infinite families, such as the Paley conference matrices, and weighing matrices constructed from ternary periodic Golay pairs. © 2018 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail addresses: [email protected] (D. Crnković), [email protected] (R. Egan), [email protected] (A. Švob). https://doi.org/10.1016/j.ffa.2018.09.002 1071-5797/© 2018 Elsevier Inc. All rights reserved.

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1. Introduction In this paper we are concerned with constructing self-orthogonal codes from weighing matrices, and self-orthogonal or Hermitian self-orthogonal codes from types of weighing matrices with entries in Fq which we will call Fq -weighing matrices. These generalize weighing matrices in that the entries are in any finite field Fq , with the understanding that a weighing matrix with non-zero entries in {±1} can be considered an F3 -weighing matrix under our convention. The conditions for existence are less restrictive than for generalized weighing matrices with non-zero entries in the multiplicative group of Fq , but are sufficient for our construction of Hermitian self-orthogonal codes over Fq where q ∈ {2, 3, 4}. We then define and construct orbit matrices from these structures, which in turn leads to further self-orthogonal and Hermitian self-orthogonal codes. The introduction of Fq -weighing matrices and use of orbit matrices in this manuscript are our main contributions, which build on the earlier work of Tonchev [19] where generalized weighing matrices are used to construct codes. We also present a construction of Fq -weighing matrices based on ternary periodic Golay sequences, and in doing so we establish a link between the theory of orbit matrices, and compression of complementary sequences, discussed by Ðoković and Kotsireas in [7]. It is well known that self-orthogonal codes have wide applications in communications (see e.g., [16]), including, for example, in secret sharing [4]. This is one of the key reasons why our interest is constructing Hermitian self-orthogonal codes, and not just in the minimum distance and error correcting properties of the codes constructed, as is often the primary motivation in coding theory. Where q = 4, we are particularly motivated by the construction of quantum-error-correcting codes (first discovered by Shor [18]) from Hermitian self-orthogonal linear F4 codes due to Calderbank et al. [3]. In particular, given a Hermitian self-orthogonal [n, k]4 code C such that no codeword in C ⊥ \ C has weight less than d, one can construct a quantum [[n, n − 2k, d]] code, (see [3, Theorems 2, 3]). An outline of some applications to quantum codes is also presented in [19, Section 4]. Self-orthogonal and Hermitian self-orthogonal codes of length up to 29 and dimension up to 6 over F3 and F4 respectively, with the largest possible minimum distance, were classified in [2], and Hermitian self-dual [18, 9] codes over F4 for lengths up to 18, were classified in [10]. The paper is outlined as follows. In the next section we provide the relevant background information. In Section 3 we formally define Fq -weighing matrices and demonstrate their usefulness for constructing self-orthogonal codes where q ∈ {2, 3, 4}. Following this we describe our construction using orbit matrices of weighing matrices (Section 4) and then generalize to Fq -weighing matrices (Section 5), giving examples to demonstrate the construction in each case. We conclude with our construction of Fq -weighing matrices via complementary sequences. The codes constructed in this paper have been constructed and examined using Magma [1]. Minimum distances are compared to known codes and bounds at [9].

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2. Preliminaries We begin with the required definitions and notation. We adhere mostly to the texts [6,11], to which we refer for background reading. Let A be a non-empty set, containing multiplicative identity 1 and not containing 0, such that for any a ∈ A, there is a−1 ∈ A. We let ∗ be the transposition acting on A ∪ {0} such that a∗ = a−1 and 0∗ = 0. When M = [mij ] is an n × n matrix with entries in A ∪ {0}, we let M ∗ = [m∗ji ]. If ∗ v = [v0 , . . . , vn−1 ] is a vector, then v ∗ = [v0∗ , . . . , vn−1 ]. Let In , Jn and On denote the n × n identity matrix, all ones matrix and all zeros matrix respectively, and let G be a finite group. An n × n matrix W with entries in λ {0} ∪ G such that W W ∗ = mIn + |G| G(Jn − In ) over the group ring Z[G] is a balanced generalized weighing matrix BGW(n, m, λ; G). If W is not necessarily balanced, i.e., we have just that W W ∗ = mIn over Z[G]/ZG, then W is a generalized weighing matrix GW(n, m; G). Where G is the multiplicative group of the finite field Fq , then W is a GW(n, m; Fq ) over Fq and W W ∗ = mIn . If W has entries in {0, ±1} and W W ∗ = mIn over the integers, W is a weighing matrix W(n, m), and if m = n, W is a Hadamard matrix H(n). A q-ary linear code C of length n and dimension k for a prime power q, is a k-dimensional subspace of a vector space Fnq . Elements of C are called codewords. Let x = (x1 , ..., xn ) and y = (y1 , ..., yn ) ∈ Fnq . The Hamming distance between words x and y is the number d(x, y) = |{i : xi = yi }|. The minimum distance of the code C is defined by d = min{d(x, y) : x, y ∈ C, x = y}. The weight of a codeword x is w(x) = d(x, 0) = |{i : xi = 0}|. For a linear code, d = min{w(x) : x ∈ C, x = 0}. A q-ary linear code of length n, dimension k, and distance d is called a [n, k, d]q code. We may use the notation [n, k] if the parameters d and q are unspecified. An [n, k] linear code C is said to be a best known linear [n, k] code if C has the highest minimum weight among all known [n, k] linear codes. It is said to be optimal if the minimum weight of C has the largest minimum weight among all linear [n, k] codes, and near-optimal if its minimum weight is at most 1 less than the largest possible value. The dual code C ⊥ is the orthogonal complement under the standard inner product · , ·, i.e. C ⊥ = {v ∈ Fnq : v, c = 0 for all c ∈ C}. Analogously, the Hermitian dual code C H is the orthogonal complement under the Hermitian inner product. A code C is self-orthogonal if C ⊆ C ⊥ and self-dual if C = C ⊥ . It is Hermitian self-orthogonal if C ⊆ C H and Hermitian self-dual if C = C H . 3. Orthogonality over finite fields Let q = pr for a prime p, and r ≥ 1. We define the inverse dot product on Fnq by u, vI = u · v ∗ . Note that for q ∈ {2, 3} this is just the standard inner product. This also coincides with the standard definition for Hermitian inner product where q = 4, see e.g., [11]. It also corresponds to what is termed the classical Hermitian inner product

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in [3]. For larger fields of square order, many sources (see e.g., [14, Exercise 4.9]) define the Hermitian inner product on Fnq2 is to be u, vH = u · v q , and under this definition the notion of a Hermitian self-orthogonal code over Fq2 follows. Lemma 3.1. The inverse dot product is additive if and only if q ∈ {2, 3, 4}. Proof. The statement is easily verified when q ∈ {2, 3, 4}. Suppose then that q > 4 and the x is a multiplicative generator of Fq \ {0}. It suffices to assume that n = 1. We have that 1, 1I + 1, xI = 1, 1 + xI if and only if 1 + x−1 = (1 + x)−1 . But (1 + x−1 )(1 + x) = 1 ⇒ x + x−1 + 1 = 0 ⇒ x2 + 1 + x = 0 ⇒ x2 = x−1 , a contradiction. 2 In this paper, we consider n × n matrices with entries in a finite field Fq where q ≤ 4, with the requirement that the inverse dot product of two distinct rows is zero, i.e., that for an n × n matrix W of weight m, we have W W ∗ = mIn . Under appropriate conditions these matrices generate self-orthogonal and Hermitian self-orthogonal codes. We will call such a matrix W an Fq -weighing matrix, W(n, m; Fq ). Note that this includes weighing matrices as a special case, where entries equal to −1 are replaced with 2 and q = 3. The converse does not necessarily hold. A GW(n, m; F4 ) is also a special case, as we do not require that over the multiplicative group g = G ∼ = C3 of non-zero elements of F4 that W W ∗ = mIn over Z[G]/ZG. As a demonstrative example, let F4 = {0, 1, x, x2 } and let ⎡

1 ⎢ 1 ⎢ W =⎢ 2 ⎣x x

1 1 x2 x

1 x 1 x2

⎤ 1 x ⎥ ⎥ ⎥. 1 ⎦ x2

Then W W ∗ = 4I4 = O4 , and W is a W(4, 4; F4 ) but it is known there is no GW(4, 4; C3 ). Conversely, every GW(n, m; C3 ) corresponds to a W(n, m; F4 ). Define the map f : {0} ∪ G → F4 by f (g i ) = xi , and f (0) = 0. Further, where M = [mij ] is a {0} ∪ G-matrix, let f (M ) = [f (mij )]. Lemma 3.2. If W is a GW(n, m; C3 ) then f (W ) is a W(n, m; F4 ). Proof. This is a routine exercise.

2

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For the remainder of this section let q ∈ {2, 3, 4}, and when referring to codes hereafter we write (Hermitian) self-orthogonal to indicate that it is self-orthogonal over F2 and F3 , and Hermitian self-orthogonal over F4 . Theorem 3.3. If W is a W(n, m; Fq ), where p divides m, then W generates a (Hermitian) self-orthogonal linear code over Fq . Proof. Fq is of characteristic dividing the weight of W and thus v, v = 0 for any row v of W . By definition u, v = 0 for any distinct rows u and v. The claim then follows from Lemma 3.1. 2 Corollary 3.4. If W is a W(n, m), where 3 divides m, then W generates a self-orthogonal code over F3 . Proof. Let q = 3 and replace entries equal to −1 with 2. Theorem 3.3 then gives the result. 2 4. Orbit matrices of weighing matrices Let M be an n × n matrix with entries in some set X. A permutation automorphism of M is a pair of n × n permutation matrices (P, Q) such that P M Q = M . The set of all such pairs form the permutation automorphism group of M , denoted PAut(M ) under the composition (P1 , Q1 )(P2 , Q2 ) = (P1 P2 , Q1 Q2 ). Any subgroup of the permutation automorphism group G ≤ PAut(M ) acts on rows and columns of M . If M is a weighing matrix, G acts in t orbits on both rows and columns, see e.g., [5, Theorem 2.3]. Let W be a weighing matrix and denote the G-orbits on rows and columns of W by R1 , . . . , Rt and C1 , . . . , Ct , respectively, and put |Ri | = Ωi and |Ci | = ωi , i = 1, . . . , t. Let Wij be the submatrix of W consisting of the rows belonging to the row orbit Ri and the column belonging to Cj . We denote by Γij and γij the sum of a row and column of Wij , respectively. The sums Γij and γij are well-defined, i.e. they do not depend on the choice of the row and the column, because the sums of entries of any two rows (or columns) of Wij are equal. The t × t matrix R = [Γij ] is called a row orbit matrix of W with respect to G. The t × t matrix C = [γij ] is called a column orbit matrix of W with respect to G. The following results generalize Theorem 2.6, Theorem 2.7 and Corollary 2.8 of [5], which applied to Hadamard matrices. We omit the proofs here. Theorem 4.1. Let W be a W(n, m) and G be a subgroup of the permutation automorphism group of W acting with all orbits of the same length w. Further, let R be the row orbit matrix of W with respect to G. If p is a prime dividing m, and q = pr is a prime power, then the linear code spanned by the matrix R over the field Fq is a self-orthogonal code of length t.

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Theorem 4.2. Let W be a W(n, m), G be a subgroup of the permutation automorphism group of W , and R the corresponding row orbit matrix. Further, let ωj , j = 1, . . . , t, be the lengths of the G-orbits on columns of W , and w ∈ {ωj | j = 1, . . . , t}. Let q = pr be a prime power, where p is a prime dividing m, and let the lengths of the column G-orbits of H have a property that pωj |w if ωj < w, and pw|ωj if w < ωj . Then the submatrix of R corresponding to row orbits and column orbits of length w spans a self-orthogonal code over Fq . The submatrix of an orbit matrix R corresponding to the fixed rows and fixed columns is called the fixed part of the orbit matrix R. The submatrix of R corresponding to the orbits of rows and columns of lengths greater than 1 is called the non-fixed part of the orbit matrix R. As a direct consequence of Theorem 4.2 we have the following corollary. Corollary 4.3. Let W be a W(n, m), G be a subgroup of the permutation automorphism group of W , and R the corresponding row orbit matrix. Further, let ωj , j = 1, . . . , t, be the lengths of the G-orbits on columns of W , and p be a prime that divides ωj if ωj > 1. Then the rows of the fixed part of R span a self-orthogonal code over the field Fq , where q = pr . 4.1. Codes from orbit matrices of weighing matrices A conference matrix of order n is a weighing matrix W (n, n − 1) with zeros on the main diagonal. If q ≡ 1 mod 4 is a prime power, then there is a symmetric conference matrix of order n = q + 1. For more information on conference matrices we refer the reader to [12,17]. In order to demonstrate the method for constructing self-orthogonal codes developed in this paper, we list Fp codes constructed from orbit matrices of symmetric conference matrices of order n = q + 1 that are obtained by Paley’s construction. The details about symmetric conference matrices used for obtaining orbit matrices and their permutation automorphism groups from Table 1 are given in [5,13]. In Table 1 ∗ denotes that the code is best known. Remark 4.4. Some of the best known codes listed in Table 1 are optimal and some of the codes are near-optimal. The optimal codes are [10, 6, 4]5 , [10, 4, 6]5 , [12, 5, 6]5 , [12, 7, 4]5 , [36, 26, 6]3 , [16, 10, 4]3 , [18, 12, 4]3 , [13, 11, 2]3 , [10, 8, 2]3 , [25, 4, 19]5 , [25, 21, 4]5 , [12, 10, 2]5 , [8, 2, 6]5 and [8, 6, 2]5 . The near optimal codes are [8, 3, 4]5 , [8, 5, 2]5 , [40, 32, 4]3 , [20, 16, 2]3 , [16, 6, 6]3 , [18, 6, 8]3 , [16, 4, 8]3 , [16, 12, 2]3 , [20, 14, 4]5 and [12, 2, 9]5 . In Table 2 we list Fp codes constructed from orbit matrices of a conference matrix of order 442 that was constructed via Mathon’s construction presented in [15]. All the

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Table 1 Codes constructed from non-fixed parts of orbit matrices. q 25 25 25 81 81 81 81 81 81 81 81 81 125 125 125 125 125 125

G ≤ PAut(W ) Z2 Z2 Z3 Z2 Z2 Z3 Z4 Z4 Z4 Z5 Z6 Z8 Z2 Z3 Z5 Z6 Z10 Z15

C [10, 6, 4]5 * [12, 5, 6]5 * [8, 3, 4]5 [36, 10, 16]3 * [40, 8, 20]3 [27, 5, 15]3 [20, 4, 10]3 [16, 6, 6]3 [18, 6, 8]3 [16, 4, 8]3 [13, 2, 7]3 [10, 2, 5]3 [62, 14, 31]5 * [40, 11, 20]5 * [25, 4, 19]5 * [20, 6, 10]5 [12, 2, 9]5 [8, 2, 6]5 *

C⊥ [10, 4, 6]5 * [12, 7, 4]5 * [8, 5, 2]5 [36, 26, 6]3 * [40, 32, 4]3 [27, 22, 3]3 [20, 16, 2]3 [16, 10, 4]3 * [18, 12, 4]3 * [16, 12, 2]3 [13, 11, 2]3 * [10, 8, 2]3 * [62, 48, 8]5 * [40, 29, 6]5 [25, 21, 4]5 * [20, 14, 4]5 [12, 10, 2]5 * [8, 6, 2]5 *

|Aut(C)| 480 576 1536 2880 640 2592 8 64 48 16384 207360 115200 1488 480 4800 16 41472 512

Table 2 Self-orthogonal codes constructed from non-fixed parts of orbit matrices. G ≤ PAut(W ) Z3 Z3 Z3 Z3 Z9

C [147, 73]3 [147, 73]3 [147, 73]3 [144, 72]3 [48, 24, 9]3

C⊥ [147, 74]3 [147, 74]3 [147, 74]3 [144, 72]3 [48, 24, 9]3

obtained codes are self-orthogonal. The codes with parameters [144, 72]3 and [48, 24, 9]3 are self-dual. 5. Orbit matrices of Fq -weighing matrices Let q = pr for some prime p and r ≥ 1. We now extend the definition of orbit matrices to Fq -weighing matrices. Let W be a W(n, m; Fq ) and let G ≤ PAut(W ) acting in t orbits on the set of rows and columns of W . We index the rows and columns of W with the integers 1, . . . , n and write W = [wxy ]1≤x,y≤n . Where R1 , . . . , Rt and C1 , . . . , Ct denote orbits of rows and columns of W , we define Ωi and ωi as before with the extra condition that we take their value modulo p, the characteristic of Fq . Let Wij be the submatrix of W consisting of the rows belonging to the row orbit Ri and the column belonging to the column orbit Cj . We denote by Γij and γij the sum of a row and column of Wij , respectively. The sums Γij and γij are again well-defined. The t × t matrices R = [Γij ] and C = [γij ] are the row and column orbit matrices of W with respect to G.

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Lemma 5.1. Adhering to the notation above, t 

∗ Γij γsj = δis m,

j=1

where δis is the Kronecker delta. Proof. Let x be a row from the row orbit Ri , and y be a column from the column orbit Cj . Then t 

∗ Γij γsj =

j=1

t   j=1

=

wxz

z∈Cj



t   t   

 ∗ ∗ ∗ wry = wxz wry = wxz wrz j=1 z∈Cj r∈Rs

r∈Rs

t   

t   

∗ wxz wrz =

j=1 r∈Rs z∈Cj

∗ wxz wrz =

j=1 z∈Cj r∈Rs n  

∗ wxz wrz .

r∈Rs z=1

r∈Rs j=1 z∈Cj

If i = s, then n  

∗ wxz wrz =

r∈Rs z=1



0 = 0.

r∈Rs

If i = s, then n  

∗ wxz wrz = (Ωs − 1)0 + m = m,

r∈Rs z=1

where Ωs is the length of the orbit Rs . 2 Theorem 5.2. Adhering to the notation above, t 

ωj Ω∗s Γij Γ∗sj = δis m,

j=1

where δis is the Kronecker delta. Proof. The sum of entries of the submatrix Wsj is Ωs Γsj . On the other hand, this sum is equal to ωj γsj , so γsj = Ωs ωj∗ Γsj . 2 For the remainder of this section, let q ∈ {2, 3, 4}. In Theorems 5.3 and 5.4 we show that under some conditions orbit matrices of weighing matrices, or their submatrices, span self-orthogonal codes.

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Theorem 5.3. Let W be a W(n, m; Fq ) and G be a subgroup of the permutation automorphism group of W acting with all orbits of the same length w, such that p divides m. Let R be the row orbit matrix of W with respect to G. Then R is an Fq -weighing matrix and the linear code spanned by the matrix R is a (Hermitian) self-orthogonal code of length n/w. Proof. By Theorem 5.2 we have t 

Ωs ωj∗ Γij Γ∗sj =

j=1

t 

ww∗ Γij Γ∗sj =

t 

j=1

Γij Γ∗sj = δis m.

j=1

Lemma 3.1 completes the proof. 2 Theorem 5.4. Let W be a W(n, m; Fq ) where m is divisible by p, G be a subgroup of the permutation automorphism group of W , and R be the corresponding row orbit matrix. Further, let ωj , j = 1, . . . , t, be the lengths of the G-orbits on columns of W , and suppose that ωj = 1 or ωj is divisible by p for all j. Then the submatrix of R corresponding to row orbits and column orbits of length w ∈ {ωj } spans a (Hermitian) self-orthogonal code over Fq , if the number of non-zero entries in each row of the submatrix is divisible by p. Proof. We show that distinct rows are pairwise orthogonal, and apply Lemma 3.1. Let the ith and the sth row orbit have length w, i.e. Ωi = Ωs = w. Then t 



ωj Ω∗s Γij Γ∗sj =

j=1

ωj Ω∗s Γij Γ∗sj +

=

ωj Ω∗s Γij Γ∗sj

j, ωj =w

j, ωj =w







ww∗ Γij Γ∗sj +

ωj w∗ Γij Γ∗sj .

j, ωj =w

j, ωj =w

Therefore,  j, ωj =w

Γij Γ∗sj =

t 

ωj Ω∗s Γij Γ∗sj −



ωj w∗ Γij Γ∗sj

j, ωj =w

j=1

= δis m −



ωj w∗ Γij Γ∗sj .

j, ωj =w

If w = 1 then ωj is divisible by p. Otherwise w∗ is divisible by p. Hence,  j, ωj =w

Γij Γ∗sj ≡ 0 mod p. 2

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We define the fixed part and non-fixed part of an orbit matrix R as before. As a direct consequence of Theorem 5.4 we have the following corollary. Corollary 5.5. Let R be the row orbit matrix constructed in Theorem 5.4. Then the rows of the fixed part of R span a (Hermitian) self-orthogonal code over Fq , if the number of non-zero entries in the fixed part of R is divisible by p. Example 5.6. Let H be the regular Hadamard matrix of order 36 below. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 − − − − − − − − − − − − − − − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

− 1 1 1 1 1 − − 1 − 1 − 1 − 1 − 1 − 1 − 1 1 1 − 1 − 1 − 1 − 1 1 1 − 1 −

− 1 1 1 1 1 − − 1 1 − 1 − − − 1 − 1 1 1 − 1 1 − − 1 − 1 1 1 − 1 1 − − 1

− 1 1 1 1 1 1 1 − − − − − − 1 1 1 1 1 − − − − 1 1 1 1 1 1 − − − − 1 1 1

− 1 1 1 1 1 − 1 − − 1 1 − 1 − − − 1 − 1 1 1 − 1 1 − − 1 − 1 1 1 − 1 1 −

− 1 1 1 1 1 1 − − 1 − − 1 1 − − 1 − − 1 1 − 1 1 − 1 1 − − 1 1 − 1 1 − 1

− − 1 − 1 − 1 1 1 1 1 − − 1 − 1 1 − 1 − 1 1 1 − 1 − 1 1 − 1 − 1 − 1 − 1

− 1 − − − 1 1 1 1 1 1 − − 1 1 − − 1 1 1 − 1 1 − − 1 1 1 − − 1 − 1 1 1 −

− − − − 1 1 1 1 1 1 1 1 1 − − − 1 1 1 − − − − 1 1 1 − − 1 1 1 1 1 1 − −

− 1 − 1 − − 1 1 1 1 1 − 1 − − 1 − 1 − 1 1 1 − 1 1 − 1 − 1 1 − − 1 − 1 1

− − 1 1 − − 1 1 1 1 1 1 − − 1 − 1 − − 1 1 − 1 1 − 1 − 1 1 − 1 1 − − 1 1

− − − 1 − 1 − 1 − 1 − 1 1 1 1 1 1 − 1 − 1 1 − 1 − 1 1 1 − 1 − 1 1 − 1 −

− − − 1 1 − 1 − − − 1 1 1 1 1 1 − 1 1 1 − − 1 1 1 − 1 1 − − 1 1 1 − − 1

− 1 1 − − − − − − 1 1 1 1 1 1 1 1 1 1 − − 1 1 1 − − − − 1 1 1 − − 1 1 1

− − 1 − − 1 1 − 1 − − 1 1 1 1 1 − 1 − 1 1 − 1 − 1 1 1 − 1 1 − 1 − 1 1 −

− 1 − − 1 − − 1 1 − − 1 1 1 1 1 1 − − 1 1 1 − − 1 1 − 1 1 − 1 − 1 1 − 1

1 1 1 − 1 − 1 1 − 1 − 1 1 − 1 − 1 1 1 1 1 − 1 − 1 − − 1 − 1 − − 1 − 1 −

1 1 1 − − 1 1 1 − − 1 1 1 − − 1 1 1 1 1 1 1 − − − 1 1 − − − 1 1 − − − 1

1 − − 1 1 1 − − 1 1 1 − − 1 1 1 1 1 1 1 1 − − − 1 1 − − − 1 1 − − − 1 1

1 1 − 1 1 − 1 − 1 1 − 1 − 1 1 − 1 1 1 1 1 1 − 1 − − 1 − 1 − − 1 − 1 − −

1 − 1 1 − 1 − 1 1 − 1 − 1 1 − 1 1 1 1 1 1 − 1 1 − − − 1 1 − − − 1 1 − −

1 1 − 1 − 1 1 − 1 − 1 1 1 − 1 − − − 1 − 1 1 1 1 1 1 − 1 − 1 − − − 1 − 1

1 − 1 1 1 − − 1 1 1 − 1 1 − − 1 − − 1 1 − 1 1 1 1 1 1 − − − 1 − − 1 1 −

1 1 1 1 − − 1 1 1 − − − − 1 1 1 1 1 − − − 1 1 1 1 1 − − − 1 1 1 1 − − −

1 − 1 − 1 1 − 1 − 1 1 1 − 1 1 − − 1 − − 1 1 1 1 1 1 1 − 1 − − − 1 − − 1

1 1 − − 1 1 1 − − 1 1 − 1 1 − 1 1 − − 1 − 1 1 1 1 1 − 1 1 − − 1 − − 1 −

1 1 1 − 1 − 1 − 1 − 1 1 − 1 − 1 − − 1 − 1 − − 1 − 1 1 1 1 1 1 − 1 − 1 −

1 1 1 − − 1 − 1 1 1 − − 1 1 1 − − − 1 1 − − − 1 1 − 1 1 1 1 1 1 − − − 1

1 − − 1 1 1 1 1 1 − − 1 1 1 − − 1 1 − − − 1 1 − − − 1 1 1 1 1 − − − 1 1

1 1 − 1 1 − − 1 − 1 1 − 1 − 1 1 − 1 − − 1 − 1 − − 1 1 1 1 1 1 1 − 1 − −

1 − 1 1 − 1 1 − − 1 1 1 − − 1 1 1 − − 1 − 1 − − 1 − 1 1 1 1 1 − 1 1 − −

1 1 − 1 − 1 1 1 − 1 − 1 − 1 − 1 − − 1 − 1 − 1 − 1 − − − 1 − 1 1 1 1 1 1

1 − 1 1 1 − 1 1 − − 1 − 1 1 1 − − − 1 1 − 1 − − − 1 − − 1 1 − 1 1 1 1 1

1 1 1 1 − − − − 1 1 1 1 1 1 − − 1 1 − − − − − − 1 1 1 1 − − − 1 1 1 1 1

1 − 1 − 1 1 1 − 1 1 − − 1 − 1 1 − 1 − − 1 1 − 1 − − − 1 − − 1 1 1 1 1 1

1 1 − − 1 1 − 1 1 − 1 1 − − 1 1 1 − − 1 − − 1 1 − − 1 − − 1 − 1 1 1 1 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Over F3 , this matrix generates a [36, 13, 12]3 self-orthogonal code. There is C3 ≤ PAut(H) acting in 10 orbits of length 3 and 6 fixed points. Over F3 , the fixed part of R is a 6 × 6 matrix of all 2s, which is trivially self-orthogonal. The submatrix of R constructed from orbits of length 3 is of weight 9 and produces a [10, 4, 6]3 (optimal) self-orthogonal code. This submatrix with −1 in place of 2 is also a W(10, 9). 5.1. Codes from orbit matrices of Fq -weighing matrices To demonstrate this approach, we obtained a W(72, 72; F4 ) from a GW(72, 72; C3 ), i.e., a generalized Hadamard matrix of order 72 over C3 , via Lemma 3.2. This matrix generates a [72, 18, 16]4 Hermitian self-orthogonal code. We note that while this code is not optimal, some of the codes obtained from its orbit matrices are. From subgroups of its permutation automorphism groups, we constructed several orbit matrices and obtained the Hermitian self-orthogonal codes, and their duals, given in Table 3. The Hermitian self-orthogonal codes come from the fixed parts, and non-fixed parts of the orbit matrices. The codes with parameters [12, 3, 8]4 , [10, 2, 8]4 and [10, 8, 2]4 are optimal.

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Table 3 Codes over F4 constructed from fixed and non-fixed parts of orbit matrices. G ≤ PAut(W )

C

C⊥

|Aut(C)|

Z2 Z2 Z2 Z2 Z4 Z4

[12, 3, 8] [30, 6, 16] [34, 8, 8] [24, 6, 8] [14, 3, 4] [10, 2, 8]

[12, 9, 2] [30, 24, 3] [34, 26, 2] [24, 18, 2] [14, 11, 2] [10, 8, 2]

2 9 · 33 · 51 2 5 · 34 · 52 2304 219 · 34 210 · 34 · 51 5760

6. Constructions from complementary sequences Define the map f : {0, ±1} → F3 by f (x) = x mod 3. Further, where M = [mij ] is a {0, ±1}-matrix, let f (M ) = [f (mij )]. Clearly if W is a W(n, m), then f (W ) is a W(n, m; F3 ). Under the conditions of Theorem 5.3, the orbit matrix R of f (W ) is also an F3 -weighing matrix. In some instances, we also have that f −1 (R) is a weighing matrix over the integers. In this section we describe a construction of weighing matrices that have this property. These matrices always have cyclic automorphism groups acting such that orbits of rows and columns are of equal length. Let a = [a0 , . . . , an−1 ] be a {0, ±1}-sequence of length n. The periodic autocorrelation n−1 function of a for a given shift s is defined to be PAFs (a) = i=0 ai ai+s where the sequence indices are read modulo n. A pair (a, b) of {0, ±1}-sequences is a ternary periodic Golay pair if PAFs (a) + PAFs (b) = 0 for all 1 ≤ s ≤ n − 1. Let A and B be the circulant matrices with first rows a and b respectively. Then the matrix W =

A −B 

B A

is a weighing matrix. In the case that both a and b contain no zero entries, (a, b) is a periodic Golay pair and W is Hadamard. The literature on periodic Golay pairs is reasonably extensive, we refer the reader to [7] for recent progress, and to the references contained therein for further background. By construction, the cyclic group Cn is a subgroup of the permutation automorphism group of W , acting cyclically on the first and second n rows and columns giving two orbits of length n. Thus for any divisor k of n, there is a cyclic group acting on rows and columns of W giving exactly 2n k orbits of length k. This group action on rows and columns is equivalent to shifting the sequences a and b cyclically prior to constructing the matrix W . Let  = nk . k−1 Let a(k) = [ i=0 ai+j ]0≤j≤−1 , where entries are taken modulo 3 and then entries equal to 2 are replaced with −1. Lemma 6.1. Let (a, b) be a ternary periodic Golay pair of length n. For any k dividing n, the sequences a(k) and b(k) comprise a ternary periodic Golay pair of length . Proof. For any 1 ≤ s ≤  − 1, PAFs (a(k) ) =

k−1 i=0

PAFi+s (a).

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Remark 6.2. Constructing the sequences a(k) and b(k) is essentially what is referred to as m-compression of periodic Golay pairs in [7], working modulo 3. Let W (k) denote the weighing matrix constructed from the ternary periodic Golay pair (a(k) , b(k) ). Theorem 6.3. Let R be the orbit matrix corresponding to the matrix W obtained from a subgroup of the permutation automorphism group Ck , where the entries are taken modulo 3 and then entries equal to 2 are replaced with −1. Then R = W (k) . We conclude this section with a complete example. To ease notation we write − for −1. Let a = [1, −, −, 1, −, 1, −, −, −, 1] and b = [1, −, −, −, −, −, −, 1, 1, −], and observe that (a, b) is a periodic Golay pair. Then the matrix ⎡

1 ⎢ 1 ⎢ ⎢ ⎢− ⎢ ⎢− ⎢ ⎢− ⎢ ⎢ ⎢1 ⎢ ⎢− ⎢ ⎢1 ⎢ ⎢− ⎢ ⎢− ⎢ W =⎢ ⎢− ⎢ ⎢1 ⎢ ⎢1 ⎢ ⎢ ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢ ⎢− ⎢ ⎢ ⎣− 1

− 1 1 − − − 1 − 1 − 1 − 1 1 1 1 1 1 − −

− − 1 1 − − − 1 − 1 − 1 − 1 1 1 1 1 1 −

1 − − 1 1 − − − 1 − − − 1 − 1 1 1 1 1 1

− 1 − − 1 1 − − − 1 1 − − 1 − 1 1 1 1 1

1 − 1 − − 1 1 − − − 1 1 − − 1 − 1 1 1 1

− 1 − 1 − − 1 1 − − 1 1 1 − − 1 − 1 1 1

− − 1 − 1 − − 1 1 − 1 1 1 1 − − 1 − 1 1

− − − 1 − 1 − − 1 1 1 1 1 1 1 − − 1 − 1

1 − − − 1 − 1 − − 1 1 1 1 1 1 1 − − 1 −

1 − 1 1 − − − − − − 1 − − 1 − 1 − − − 1

− 1 − 1 1 − − − − − 1 1 − − 1 − 1 − − −

− − 1 − 1 1 − − − − − 1 1 − − 1 − 1 − −

− − − 1 − 1 1 − − − − − 1 1 − − 1 − 1 −

− − − − 1 − 1 1 − − − − − 1 1 − − 1 − 1

− − − − − 1 − 1 1 − 1 − − − 1 1 − − 1 −

− − − − − − 1 − 1 1 − 1 − − − 1 1 − − 1

1 − − − − − − 1 − 1 1 − 1 − − − 1 1 − −

1 1 − − − − − − 1 − − 1 − 1 − − − 1 1 −

⎤ − ⎥ 1⎥ ⎥ 1⎥ ⎥ −⎥ ⎥ −⎥ ⎥ ⎥ −⎥ ⎥ −⎥ ⎥ −⎥ ⎥ −⎥ ⎥ 1⎥ ⎥ ⎥ −⎥ ⎥ −⎥ ⎥ 1⎥ ⎥ ⎥ −⎥ ⎥ 1⎥ ⎥ −⎥ ⎥ −⎥ ⎥ −⎥ ⎥ ⎥ 1⎦ 1

is Hadamard. Let P be the circulant matrix with first row [0, 0, 0, 0, 0, 0, 0, 0, 0, 1]. Then (I2 ⊗ P )W (I2 ⊗ P ) = W , and thus ((I2 ⊗ P ), (I2 ⊗ P )) generates a subgroup of the permutation automorphism group of W isomorphic to C10 , and orbits of rows/columns under this automorphism group are all of length 10. Letting Rk be the orbit matrix corresponding to Ck  C10 where entries are take modulo 3 and 2 is replaced with −, we get the following:

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⎡

− ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢1 ⎢ ⎢1 R2 = ⎢ ⎢0 ⎢ ⎢ ⎢− ⎢ ⎢0 ⎢ ⎣− −

1 − 0 0 1 − 0 − 0 −

1 1 − 0 0 0 − 0 − 0

0 1 1 − 0 0 0 − 0 −

0 0 1 1 − − 0 0 − 0

0 1 0 0 1 − 1 1 0 0

1 0 1 0 0 0 − 1 1 0

0 1 0 1 0 0 0 − 1 1

0 0 1 0 1 1 0 0 − 1

⎤ 1 0⎥ ⎥ 0⎥ ⎥ ⎡ ⎥ 0 1⎥ ⎥ ⎢ ⎥ 0⎥ ⎢1 , R5 = ⎢ ⎣1 1⎥ ⎥ ⎥ 1⎥ 0 ⎥ 0⎥ ⎥ 0⎦ −

1 0 0 1

− 0 0 1

⎤ 0 −⎥ 1 ⎥ ⎥ , R10 = 1⎦ 1 0

− 1

Finally observe that (a(2) , b(2) ) = ([−, 1, 1, 0, 0], [0, 1, 0, 0, 1]), (a(5) , b(5) ) = ([0, 1], [−, 0]) and (a(10) , b(10) ) = ([1], [−]), and thus W (k) = Rk for all k = 2, 5, 10. 6.1. Constructing F4 -weighing matrices The notion of complementary sequences may be generalized to sequences with entries in Fq . In this section we demonstrate constructing Fq -weighing matrices from complementary sequences, which is a minor adaptation of the construction of Butson Hadamard matrices outlined in [8]. For our purposes we let q = 4 in this section, and let F4 = {0, 1, x, x2 }. We define the periodic autocorrelation function of an F4 -sequence a n−1 of length n for a given shift s to be PAFs (a) = i=0 ai a∗i+s where the sequence indices are read modulo n. Under this definition, a pair (a, b) of F4 -sequences is an F4 -periodic Golay pair if PAFs (a) + PAFs (b) = 0 for all 1 ≤ s ≤ n − 1. Let A and B be the circulant matrices with first rows a and b respectively. Then the matrix W =

A B∗

B A∗

is an F4 -weighing matrix. The cyclic group Cn acts on W in exactly the same way as before, and orbit matrices over F4 are thus constructed. Linear F4 codes generated by W , or by the orbit matrices of W with respect to some G ≤ Cn , are Hermitian self-orthogonal. A further exposition into this construction is beyond the scope of this paper, however we conclude with a small example to demonstrate that this construction is not vacuous. Observe that the sequences [x, 0, x, x, 1, 0] and [x2 , x, 1, 1, 1, 1] form an F4 -periodic Golay pair of length 6. The matrix constructed as above generates a Hermitian self-orthogonal quasicyclic [12, 5, 4]4 linear code. The orbit matrix R with respect to the subgroup of the permutation automorphism group C3 generates a Hermitian self-orthogonal quasicyclic [4, 2, 2]4 linear code.

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