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The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

Constructing locally repairable codes achieving Singleton-type bound (we call them optimal codes in this paper) is a challenging task and has attracted great attention in the last few years. Tamo and Barg first gave a breakthrough result in this topic by cleverly considering subcodes of Reed-Solomon codes. Thus, $q$ -ary optimal locally repairable codes from subcodes of Reed-Solomon codes given by Tamo and Barg have length upper bounded by $q$. Recently, it was shown through extension of construction by Tamo and Barg that length of $q$ -ary optimal locally repairable codes can be $q+1$ by Jin et al.. Surprisingly it was shown by Barg et al. that, unlike classical MDS codes, $q$ -ary optimal locally repairable codes could have length bigger than $q+1$. Thus, it becomes an interesting and challenging problem to construct $q$ -ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we make use of rich algebraic structures of elliptic curves to construct a family of $q$ -ary optimal locally repairable codes of length up to $q+2\sqrt {q}$. It turns out that locality of our codes can be as big as 23 and distance can be linear in length. [ABSTRACT FROM AUTHOR]

Locally repairable codes with more than one recovering set are demanded in the application to distributed storage. For each failure node (or disk), it is desired to have as many recovering sets as possible. In this paper, we make use of automorphisms of rational function fields to construct locally repairable codes with multiple recovering sets. Although we focus on two recovering sets, our construction can be easily generalized to the case of multiple recovering sets. In particular, we obtain a class of locally repairable codes with minimum distance only 1 less than the upper bound. [ABSTRACT FROM AUTHOR]

In this paper, we determine all dihedral subgroups in $PGL_{2}(\mathbb {F}_{q})$ , where $q=2^{l}$ and $l$ is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes. [ABSTRACT FROM AUTHOR]

A Z2Z4 -linear Hadamard code of length \alpha +2\beta =2^t is a binary Hadamard code, which is the Gray map image of a Z2Z4 -additive code with \alpha binary coordinates and \beta quaternary coordinates. It is known that there are exactly \lfloor \frac t-12 \rfloor and \lfloor \frac t2 \rfloor nonequivalent Z2Z4 -linear Hadamard codes of length 2^t , with $\alpha =0$ and \alpha \not =0$ , respectively, for all t\geq 3$ . In this paper, it is shown that each \alpha =0$ is equivalent to a Z2Z4 -linear Hadamard code with \alpha \not =0 nonequivalent Z2Z4 -linear Hadamard codes of length 2^t . Moreover, the order of the monomial automorphism group for the Z2Z4 -additive Hadamard codes and the permutation automorphism group of the corresponding Z2Z4 -linear Hadamard codes are given. [ABSTRACT FROM AUTHOR]

Given a 1-perfect code C, the group of symmetries of C, Sym(C)=\\pi \in \cal Sn \,\vert \, \pi (C)=C\, is a subgroup of the group of automorphisms of C. In this paper, we focus on symmetries of order two, i.e., involutions. Let InvF(C) \subseteq Sym(C) be the set of involutions that stabilize F pointwise. For linear 1-perfect codes, the possibilities for the number of fixed points \vert F\vert are given, establishing lower and upper bounds. For any m \geq 2 and any value k between these bounds, \lceil m\over 2\rceil \leq k \leq m-1, linear 1-perfect codes of length n=2^m-1 which have an involution that fixes \vert F\vert =2^k-1 coordinates are constructed. Moreover, for any m\geq 4, 1 \leq r \leq m-1, and \lceil m\over 2\rceil \leq k \leq m-1, nonlinear 1-perfect codes of length n=2^m-1 having rank n-m+r and an involution that fixes 2^k-1 coordinates are also constructed, except one case, when m \geq 6 is even, r=m-1 and k= m\over 2. [ABSTRACT FROM AUTHOR]

This paper describes scale-recursive symbol constellations for the synchronous multiple-access (MA) channel, which can be seen as a generalization of Voronoi coding. The constellations are related by a fixed scaling and rotation, and are designed so that in the noise-free case, the combined signal always falls onto an appropriate lattice. This allows high throughput and a linear complexity decoding algorithm. Novel constellations of up to 24 dimensions are given that show improved performance over conventional rectangular lattices. [ABSTRACT FROM AUTHOR]

Given a 1-perfect code C, the group of symmetries of C, Sym(C)=\\pi\in Sn\,\vert\,\pi (C)=C\, is a subgroup of the group of automorphisms of C. In this paper, we focus on symmetries of order 2, i.e., involutions. Let Inv(C)\subseteq Sym(C) be the set of involutions of C. An additive code C of type (\alpha,\beta) is a subgroup of \BBZ2^{\alpha}\times\BBZ4^{\beta}. For additive 1-perfect codes C of length 2^t-1, we establish that any involution \pi\in Inv(C) must have at least 2^k-1 fixed points where k\geq\lceilt\over2\rceil. [ABSTRACT FROM AUTHOR]

Perfect codes obtained by the Vasil'ev–Schönheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least \exp (cN^2) propelinear 1-perfect codes of length N over an arbitrary finite field, while an upper bound on the number of transitive codes is \exp (C(N\ln N)^2^\vphantom)). [ABSTRACT FROM PUBLISHER]

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that g^p is a fixed point free involution. If C is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C(g^p) which consists of the set of fixed points of g^p, we prove that C is a projective \BBF 2\langle g \rangle -module if and only if a natural projection of C(g^p) is a self-dual code. We then discuss easy-to-handle criteria to decide if C is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58. [ABSTRACT FROM PUBLISHER]

A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 that has an automorphism group containing either the dihedral group of order 10, the elementary abelian group of order 9, or the cyclic group of order 7. Combining this with the known results in the literature, one obtains that the order of Aut(C) is either 5 or divides 24. [ABSTRACT FROM PUBLISHER]

An optical orthogonal signature pattern code (OOSPC) finds application in transmitting 2-D images through multicore fiber in code-division multiple-access (CDMA) communication systems. Observing a one-to-one correspondence between an OOSPC and a certain combinatorial subject, called a packing design, we present a construction of optimal OOSPCs with weight 4 and maximum collision parameter 2, which generalizes a well-known Köhler construction of optimal optical orthogonal codes (OOC) with weight 4 and maximum collision parameter 2. Using this new construction enables one to obtain infinitely many optimal OOSPCs, whose existence was previously unknown. We prove that for a multiple n of 4, there exists no optimal OOSPC of size 6 x n with weight 4 and maximum collision parameter 2, together with a report which shows a gap between optimal OOCs and optimal OOSPCs when 6 and n are not coprime. We also present a recursive construction of OOSPCs which are asymptotically optimal with respect to the Johnson bound. As a by-product, we obtain an asymptotically optimal (m, n, 4, 2)-OOSPC for all positive integers m and n. [ABSTRACT FROM AUTHOR]

The dual of the Kasami code of length $q^{2}-1$ , with $q$ a power of 2, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_{q}$ with a Simplex code of length $q-1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely regular. The automorphism groups of the Kasami code and the related $q$ -ary MDS code are determined. New cyclic completely regular codes over finite fields a power of 2, generalized Kasami codes, are constructed; they have coset graphs isomorphic to that of the Kasami codes. Another wide class of completely regular codes, including additive codes, as well as unrestricted codes, is obtained by combining cosets of the Kasami or generalized Kasami code. [ABSTRACT FROM AUTHOR]

The existence of an extremal code of length 72 is a long-standing open problem. Let \cal C be a putative extremal code of length 72 and suppose that \cal C has an automorphism g of order 6. We show that \cal C, as an \BBF2\langle g\rangle-module, is the direct sum of two modules; it is easy to determine one of them, while the other one has a very restrictive structure. We use this fact to do an exhaustive search and we do not find an extremal code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6. [ABSTRACT FROM AUTHOR]

Affine Grassmann codes are a variant of generalized Reed–Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed–Muller codes. [ABSTRACT FROM PUBLISHER]

Best and Brouwer refid="ref2"/ proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2^m-4 and 2^m-3, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computer-aided classification of the optimal binary one-error-correcting codes of lengths 12 and 13 possible; there are 237\thinspace610 and 117\thinspace823 such codes, respectively (with 27\thinspace375 and 17\thinspace513 inequivalent extensions). This completes the classification of optimal binary one-error-correcting codes for all lengths up to 15. Some properties of the classified codes are further investigated. Finally, it is proved that for any m \geq 4, there are optimal binary one-error-correcting codes of length 2^m-4 and 2^m-3 that cannot be lengthened to perfect codes of length 2^m-1. [ABSTRACT FROM AUTHOR]

We prove that the automorphism group of a binary self-dual doubly even [72, 36, 16] code has order 5, 7, 10, 14 or d where d divides 18 or 24, or it is A4 \times C3. [ABSTRACT FROM AUTHOR]

We consider a new class of linear codes, called affine Grassmann codes. These can be viewed as a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. We determine the length, dimension, and the minimum distance of any affine Grassmann code. Moreover, we show that affine Grassmann codes have a large automorphism group and determine the number of minimum weight codewords. [ABSTRACT FROM AUTHOR]

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász's Local Lemma (LLL) and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum-likelihood (ML) decoding. [ABSTRACT FROM AUTHOR]

Some methods to determine the local weight distribution of binary linear codes are presented. Two approaches are studied: A computational approach and a theoretical approach. For the computational approach, an algorithm for computing the local weight distribution of codes using the automorphism group of the codes is devised. In this algorithm, a code is considered the set of cosets of a subcode, and the set of cosets is partitioned into equivalence classes. Thus, only the weight distributions of zero neighbors for each representative coset of equivalence classes are computed. For the theoretical approach, relations between the local weight distribution of a code, its extended code, and its even weight subcode are studied. As a result, the local weight distributions of some of the extended primitive Bose-Chaud- huri-Hocquenghen (BCH) codes, Reed-Muller codes, primitive BCH codes, punctured Reed-Muller codes, and even weight subcodes of primitive BCH codes and punctured Reed-Muller codes are determined. [ABSTRACT FROM AUTHOR]

ollowing Parts I and II, quasi-cyclic codes of given index are studied as codes over a finite polynomial ring. These latter codes are decomposed by the Chinese Remainder Theorem (CRT), or equivalently the Mattson-Solomon transform, into products of shorter codes over larger alphabets. We characterize and enumerate self-dual one-generator quasicyclic codes in that context We give an algorithm to remove some equivalent codes from that enumeration. A generalization to multigenerator codes is sketched. [ABSTRACT FROM AUTHOR]

Recently, Blackford and Ray--Chaudhuri used transform domain techniques to permutation groups of cyclic codes over Galois rings. They used the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length 2m over any subring of GR (4, m) to be affine invariant. Here, we use the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length pm over any subring of GR(Pe, m) to be affine invariant, for e = 2 with arbitrary p and for p = 2 with arbitrary e. These are used to find two new classes of affine invariant Bose--Chaudhuri--Hocquenghem (BCH) and generalized Reed--Muller (GRM) codes over Z2e for arbitrary e and a class of affine invariant BCH codes over Zp2 for arbitrary prime p. [ABSTRACT FROM AUTHOR]

We give a geometric construction of a [110, 5, 90]9-linear code admitting the Mathieu group M11 as a subgroup of its automorphism group. [ABSTRACT FROM AUTHOR]