Search

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is little result on constructive lower bound of asymptotical behavior of locally repairable codes. In this paper, we extend the construction given by Barg et al. via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show that the Gilbert–Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size $q$. [ABSTRACT FROM AUTHOR]

Complexity of sequences plays an important role in pseudorandom sequences and cryptography. In this paper, we present a construction of sequences with high nonlinear complexity from function fields. The main idea is to make use of function fields with many rational places as well as an automorphism of large order. We illustrate our construction through rational function fields and cyclotomic function fields in which there exist some automorphisms of large order. It turns out that we are able to: 1) slightly increase the length of the inversive sequence without losing nonlinear complexity and 2) obtain sequences with much larger nonlinear complexity than random sequences. [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]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. The objective of this paper is to present some sufficient and necessary conditions that linear codes and cyclic codes have one-dimensional hull. It is shown that there are no such binary or ternary cyclic codes. Based on these characterizations, some constructions of linear codes with one-dimensional hull were given by employing quadratic number fields, partial difference sets, and difference sets. We also construct cyclic codes with one-dimensional hull. Some optimal codes with one-dimensional hull are obtained. [ABSTRACT FROM AUTHOR]

Let $m$ be an arbitrary positive integer and $D_{8m}$ be the dihedral group of order $8m$ , i.e., $D_{8m}=\langle x,y\mid x^{4m}=1, y^{2}=1, yxy=x^{-1}\rangle $. Left ideals of the dihedral group algebra $\mathbb {F}_{2}[D_{8m}]$ are called binary left dihedral codes of length $8m$ , and abbreviated as binary left $D_{8m}$ -codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left $D_{8m}$ -codes. These codes make up an important class of self-dual binary $[8m,4m]$ -codes such that the dihedral group $D_{8m}$ is necessarily a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left $D_{8m}$ -codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary $[{48,24,12}]$ -code and an extremal self-dual binary $[{56,28,12}]$ -code from self-dual binary left $D_{48}$ -codes and left $D_{56}$ -codes respectively. [ABSTRACT FROM AUTHOR]

Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin–Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed–Solomon- and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia–Stichtenoth tower. [ABSTRACT FROM AUTHOR]

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field $\mathbb {F}_{2^{n}}$ for a positive integer $n$. It is often overlooked in this community that all $2^{n}+1$ rational places (including “the place at infinity”) of the rational function field over $\mathbb {F}_{2^{n}}$ form a cyclic structure under an automorphism of order $2^{n}+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $2^{n}+1$ via cyclotomic function fields over $\mathbb {F}_{2^{n}}$. Each family of our sequences has size $2^{n}-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor $. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $2^{n}+1$. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^{n}-1$ for even $n$ , our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller. [ABSTRACT FROM AUTHOR]

In this paper, we construct some class of explicit subcovers of the curve $\mathcal {X}_{n,r}$ defined over $\mathbb {F}_{q^{n}}$ by affine equation $y^{q^{n-1}}+\cdots +y^{q}+y=x^{q^{n-r}+1}-x^{q^{n}+q^{n-r}}$. These subcovers are defined over $\mathbb {F}_{q^{n}}$ by affine equation $g_{s}(y)=x^{q^{n}+q^{n-r}}-x^{q^{n-r}+1}$ , where $g_{s}(y)$ is a $q$ -polynomial of degree $q^{s}$. The Weierstrass semigroup $H(P_\infty)$ , where $P_\infty $ is the only point at infinity on such subcovers, is determined for $1 \leq s \leq 2r-n+1$ , and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained. [ABSTRACT FROM AUTHOR]

Binary primitive triple-error-correcting Bose--Chaudhuri-Hocquenghem (BCH) codes of length n = 2m - 1 have been the object of intensive studies for several decades. In the 1970s, their covering radius was determined in a series of papers to be ρ = 5. However, one problem for these codes that has been open up to now is to find their coset distribution. In this paper this problem is solved and the number of cosets of each weight in any binary primitive triple-error-correcting BCH code is determined. As a consequence this also gives the coset distribution of the extended codes of length N = 2m with minimum distance 8. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there does not exist a binary self-dual doubly even [72, 36, 16] code with an automorphism of order 9. To do so, we apply a method for constructing binary self-dual codes possessing an automorphism of order p^2 for an odd prime p. [ABSTRACT FROM PUBLISHER]

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]

In the present paper, by making use of some special properties of the Hermitian function fields, we construct multisequences with both large linear complexity and k-error linear complexity. Moreover, these sequences can be explicitly constructed. [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]

In this paper, we simplify and extend the Eta pairing, originally discovered in the setting of supersingular curves by Barreto et al., to ordinary curves. Furthermore, we show that by swapping the arguments of the Eta pairing, one obtains a very efficient algorithm resulting in a speed-up of a factor of around six over the usual Tate pairing, in the case of curves that have large security parameters, complex multiplication by an order of Q(√-3), and when the trace of Frobenius is chosen to be suitably small. Other, more minor savings are obtained for more general curves. [ABSTRACT FROM AUTHOR]

In this paper, we present some results concerning a decomposition of binary self-dual codes having an automorphism of an order which is the product of two odd prime numbers. These results are applied to construct self-dual [72, 36, 12] codes with an automorphism of order 15. Furthermore, it is proved that the automorphism group of a putative binary extremal self-dual [72, 36, 16] code contains at most 28 nontrivial types of automorphisms of odd order. A complete list of these possible types of automorphisms is presented. [ABSTRACT FROM AUTHOR]

The main practical limitation of the McEliece public-key encryption scheme is probably the size of its key. A famous trend to overcome this issue is to focus on subclasses of alternant/Goppa codes with a non-trivial automorphism group. Such codes display then symmetries allowing compact parity-check or generator matrices. For instance, a key-reduction is obtained by taking quasi-cyclic (QC) or quasi-dyadic (QD) alternant/Goppa codes. We show that the use of such symmetric alternant/Goppa codes in cryptography introduces a fundamental weakness. It is indeed possible to reduce the key-recovery on the original symmetric public-code to the key-recovery on a (much) smaller code that has no symmetry anymore. This result is obtained thanks to an operation on codes called folding that exploits the knowledge of the automorphism group. This operation consists in adding the coordinates of codewords which belong to the same orbit under the action of the automorphism group. The advantage is twofold. The reduction factor can be as large as the size of the orbits, and it preserves a fundamental property: folding the dual of an alternant (respectively, Goppa) code provides the dual of an alternant (respectively, Goppa) code. A key point is to show that all the existing constructions of alternant/Goppa codes with symmetries follow a common principal of taking codes whose support is globally invariant under the action of affine transformations (by building upon prior works of Berger and Dür). This enables not only to present a unified view but also to generalize the construction of QC, QD, and even quasi-monoidic Goppa codes. Finally, our results can be harnessed to boost up any key-recovery attack on McEliece systems based on symmetric alternant or Goppa codes, and in particular algebraic attacks. [ABSTRACT FROM PUBLISHER]

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]

Let C be a binary extremal self-dual code of length 96. We prove that an automorphism of C of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then Aut(C) is solvable and its order divides 2^53 or 2^55 or Aut(C) is the alternating group A5 which is the only possible group of order 60. Furthermore, \vert Aut(C)\vert = 20 or 40 cannot occur. [ABSTRACT FROM AUTHOR]

Presents a method of determining the automorphism group of a huge class of Goppa codes associated with certain function fields. Hyperelliptic codes in odd characteristic; Special class of function fields; Conclusion.

Using a method for constructing self-dual codes having an automorphism of odd prime order, we classify up to equivalence all binary self-dual codes with an automorphism of order 11 with 6 cycles and minimum distance 12. This classification gives new [72, 36, 12] codes with weight enumerator that was previously not obtained as well as many [66, 33, 12], [68, 34, 12], and [70, 35, 12] codes with new values of the parameters in their respective weight enumerators. [ABSTRACT FROM PUBLISHER]

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring \BBZ4 is developed. Our approach is based on the classical Brouwer–Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the \BBZ4-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all \BBZ4-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the \BBZ4-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the \BBZ4-linear XQR-code of length 80 is a new example of an extremal \BBZ4-linear type II code. Additionally, we give the symmetrized weight enumerator of the \BBZ4-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the \BBZ4-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze [ABSTRACT FROM PUBLISHER]

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]

All binary [n,n/2] optimal self-dual codes for length 52\leq n\leq 60 with an automorphism of order 7 or 13 are classified up to equivalence. Two of the constructed [54,27,10] codes have weight enumerators that were not previously known to exist. There are also some [58,29,10] codes with new values of the parameters in their weight enumerator. [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]

Let C be a binary extremal self-dual code of length n ≥ 48. We prove that for each σ ∈ Aut(C) of prime order p ≥ 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number of p-cycles. It turns out that large primes p, i.e., n - p small, seem to occur in |Aut(C)| very rarely. Examples are the extended quadratic residue codes. We further prove that doubly even extended quadratic residue codes of length n = p + 1 are extremal only in the cases n = 8, 24, 32, 48, 80, and 104. [ABSTRACT FROM AUTHOR]

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code. [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]

We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight. [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]

In this correspondence, we prove that the automorphism group of a putative binary self-dual doubly even [72, 36, 16) code is solvable. Moreover, its order is 5, 7, 10, 14, 56, or a divisor of 72. [ABSTRACT FROM AUTHOR]

In this correspondence, we show that in many cases, the automorphism group of a curve and the permutation automorphism group of a corresponding AG code are the same. This generalizes a result of Wesemeyer beyond the case of planar curves. [ABSTRACT FROM AUTHOR]

In this correspondence, we present an interleaving scheme that yields quasi-cyclic turbo codes. We prove that randomly chosen members of this family yield with probability almost 1 turbo codes with asymptotically optimum minimum distance, i.e., growing as a logarithm of the interleaver size. These interleavers are also very practical in terms of memory requirements and their decoding error probabilities for small block lengths compare favorably with previous interleaving schemes. [ABSTRACT FROM AUTHOR]

A pair of posets (P, Q) on [n] is called a weak dual Mac Williams pair (wdMp) if the P-weight enumerator of a linear code uniquely determines the Q-weight enumerator of the dual of that code for every linear code of length n over a finite field. First, we show that (P, ...) is a wdMp if and only if the group of all P-weight preserving linear automorphisms of the ambient n-dimensional space over the finite field acts transitively on every P-sphere centered at 0. Here ... is the dual poset of P. Also, we show some equivalent conditions which say that P being weak order poset with Q = ... is essentially the only possible case for(P,Q) to be a wdMp. [ABSTRACT FROM AUTHOR]

Characterizes the full automorphism groups of homogeneous and projective Reed-Muller codes. Definition and properties of the automorphisms groups of a code; Proof regarding the invariability of the codes under particular automorphisms; Permutation group of homogeneous Reed-Muller codes.

Presents a study which aimed to determine the automorphism groups of the codes from curves more general than Hermitian curves. Discussion on the main theorem; Specific divisors; Generalizations of Xing's lemmas; Proof of main theorem.

Presents information on a study which investigated the binary self-dual codes with an automorphism of order two. Results about binary self-dual codes that have an automorphism of order two; Method for constructing a binary self-dual code.

Discusses a mass formula for quaternary self-dual and type II codes for self-dual codes over Fq + microFq rings. General linear codes; Characterization of self-dual codes; Automorphisms group.

Presents a study which gave a condition for a classical Goppa code to have a cyclic extension. Semilinear automorphism group where the condition was derived; Construction of families of Goppa codes with a cyclic extension.

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]