• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-08-07
Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E. Praeger

In this paper, we first study biplanes $$\mathcal {D}$$ with parameters (v, k, 2), where the block size $$k\in \{13,16\}$$. These are the smallest parameter values for which a classification is not available. We show that if $$k=13$$, then either $$\mathcal {D}$$ is the Aschbacher biplane or its dual, or $$\mathbf {Aut}(\mathcal {D})$$ is a subgroup of the cyclic group of order 3. In the case where

更新日期：2020-08-08
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-08-06
Martin Ekerå

We revisit the quantum algorithm for computing short discrete logarithms that was recently introduced by Ekerå and Håstad. By carefully analyzing the probability distribution induced by the algorithm, we show its success probability to be higher than previously reported. Inspired by our improved understanding of the distribution, we propose an improved post-processing algorithm that is considerably

更新日期：2020-08-06
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-31
Yan Zhu, Naoki Watamura

Relative t-designs are defined in both P- and Q-polynomial association schemes. In this paper, we investigate relative t-designs in Johnson association schemes J(v, k) for P-polynomial structure. It is known that each nontrivial shell of J(v, k) is identified with the product of two smaller Johnson association schemes. We prove that relative t-designs in J(v, k) supported by one shell are equivalent

更新日期：2020-08-01
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-03
Ignacio García-Marco, Irene Márquez-Corbella, Diego Ruano

Given a linear code $${\mathcal {C}}$$, its square code $${\mathcal {C}}^{(2)}$$ is the span of all component-wise products of two elements of $${\mathcal {C}}$$. Motivated by applications in multi-party computation, our purpose with this work is to answer the following question: which families of affine variety codes have simultaneously high dimension $$k({\mathcal {C}})$$ and high minimum distance

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-08
Alain Couvreur, Isabella Panaccione

We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance. When applied to Reed–Solomon or algebraic geometry codes, the algorithm is a reformulation of the so-called power decoding algorithm. Asymptotically, it corrects errors up to Sudan’s radius. In addition, this new framework applies to any code benefiting from an error

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-06-21
Umberto Martínez-Peñas

Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least 3 (thus one-error-correcting) for a fixed redundancy r, base-field size q and field-extension degree m (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-24
Yunwen Liu, Wenying Zhang, Bing Sun, Vincent Rijmen, Guoqiang Liu, Chao Li, Shaojing Fu, Meichun Cao

For differential cryptanalysis under the single-key model, the key schedules hardly need to be exploited in constructing the characteristics, which is based on the hypothesis of stochastic equivalence. In this paper, we study a profound effect of the key schedules on the validity of the differential characteristics. Noticing the sensitivity in the probability of the characteristics to specific keys

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-17
Daniel Coggia, Alain Couvreur

We present a polynomial time attack of a rank metric code based encryption scheme due to Loidreau for some parameters.

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-03
Gretchen L. Matthews, Fernando Piñero

Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-07-02
Yanyan Gao, Qin Yue, Yansheng Wu

Let $$\mathbb {F}_q$$ be a finite field with q elements, $$D_{2n,\,r}$$ a generalized dihedral group with $$\gcd (2n,q)=1$$, and $$\mathbb {F}_q[D_{2n,\,r}]$$ a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of $$\mathbb {F}_q[D_{2n,\,r}]$$ is determined, which extends the results of Brochero Martínez (Finite Fields Appl 35:204–214, 2015). Secondly, all

更新日期：2020-07-24
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-04-27
Alonso Sepúlveda Castellanos, Maria Bras-Amorós

We determine the Weierstrass semigroup $$H(P_\infty ,P_1,\ldots ,P_m)$$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced in Tafazolian et al. (J Pure Appl Algebra 220(3):1122–1132, 2016). Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point

更新日期：2020-04-27
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-03-28
Alexandru Chirvasitu, Thomas W. Cusick

Let $$f_n(x_0, x_1, \ldots , x_{n-1})$$ denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree d in $$n \ge d$$ variables and let $$wt(f_n)$$ denote the Hamming weight of this function. Let $$(0, a_1, \ldots , a_{d-1})_n$$ denote the function $$f_n$$ of degree d in n variables generated by the monomial $$x_0x_{a_1} \ldots x_{a_{d-1}}.$$ Such a function

更新日期：2020-03-28
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-03-21
Tovohery Hajatiana Randrianarisoa

In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes. We give a complete classification of constant rank weight code and we give their generalized rank weights.

更新日期：2020-03-21
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-03-10
Irene Márquez-Corbella, Edgar Martínez-Moro, Carlos Munuera

A locally recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. In this article we develop an algorithm that computes a recovery structure as concise as possible for an arbitrary linear code $${\mathcal {C}}$$ and a recovery method that realizes it. This algorithm also provides the locality

更新日期：2020-03-10
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-02-26
Lucky Erap Galvez, Jon-Lark Kim

Matrix codes over a finite field $${\mathbb {F}}_q$$ are linear codes defined as subspaces of the vector space of $$m \times n$$ matrices over $${\mathbb {F}}_q$$. In this paper, we show how to obtain self-dual matrix codes from a self-dual matrix code of smaller size using a method we call the building-up construction. We show that every self-dual matrix code can be constructed using this building-up

更新日期：2020-02-26
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-02-11
René Bødker Christensen, Olav Geil

In this paper, we study the construction of quantum codes by applying Steane-enlargement to codes from the Hermitian function field. We cover Steane-enlargement of both usual one-point Hermitian codes and of order bound improved Hermitian codes. In particular, the paper contains two constructions of quantum codes whose parameters are described by explicit formulae, and we show that these codes compare

更新日期：2020-02-11
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2020-02-07
Hiram H. López, Gretchen L. Matthews, Ivan Soprunov

A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes, and J-affine variety codes. In this work we use the vanishing ideal of the Cartesian product to give a description of the dual of a monomial-Cartesian code. Then we use

更新日期：2020-02-07
• Des. Codes Cryptogr. (IF 1.224) Pub Date : 2019-11-20
Thomas Britz, Adam Mammoliti, Keisuke Shiromoto

We extend and provide new proofs of the Wei-type duality theorems, due to Ducoat and Ravagnani, for Gabidulin–Roth rank-metric codes and for Delsarte rank-metric codes. These results follow as corollaries from fundamental Wei-type duality theorems that we prove for certain general combinatorial structures.

更新日期：2019-11-20
• Des. Codes Cryptogr. Pub Date : null
Andries E Brouwer,Sven C Polak

For n , d , w ∈ N , let A(n, d, w) denote the maximum size of a binary code of word length n, minimum distance d and constant weight w. Schrijver recently showed using semidefinite programming that A ( 23 , 8 , 11 ) = 1288 , and the second author that A ( 22 , 8 , 11 ) = 672 and A ( 22 , 8 , 10 ) = 616 . Here we show uniqueness of the codes achieving these bounds. Let A(n, d) denote the maximum size

更新日期：2019-11-01
• Des. Codes Cryptogr. Pub Date : null
Sven C Polak

For q , n , d ∈ N , let A q ( n , d ) be the maximum size of a code C ⊆ [ q ] n with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds A 5 ( 8 , 6 ) ≤ 65 , A 4 ( 11 , 8 ) ≤ 60 and A 3 ( 16 , 11 ) ≤ 29 . These in turn imply the new upper bounds A 5 ( 9 , 6 ) ≤ 325 , A 5 ( 10 , 6 ) ≤ 1625 , A 5 ( 11 , 6 ) ≤ 8125 and A 4 ( 12 , 8 ) ≤ 240 . Furthermore, we prove

更新日期：2019-11-01
• Des. Codes Cryptogr. Pub Date : null
Arnold Neumaier

The paper describes improved analysis techniques for basis reduction that allow one to prove strong complexity bounds and reduced basis guarantees for traditional reduction algorithms and some of their variants. This is achieved by a careful exploitation of the linear equations and inequalities relating various bit sizes before and after one or more reduction steps.

更新日期：2019-11-01
• Des. Codes Cryptogr. Pub Date : null
Bart Litjens,Sven Polak,Alexander Schrijver

For nonnegative integers q, n, d, let A q ( n , d ) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on A q ( n , d ) . For any k, let C k be the collection of codes of cardinality at most k. Then A q ( n , d ) is at most the maximum value of ∑ v ∈ [ q ] n x ( { v } ) , where x is a function

更新日期：2019-11-01
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