Formally self-dual additive codes over F₄

Han, S.,Kim, J.L. Academic Press 2010 Journal of symbolic computation Vol.45 No.7

Additive codes over F<SUB>4</SUB>have been of great interest due to their application to quantum error correction. As another application, we introduce a new class of formally self-dual additive codes over F<SUB>4</SUB> which is a natural analogue of the binary formally self-dual codes and is missing in the study of additive codes over F<SUB>4</SUB> In fact, Gulliver and Ostergard (2003) considered formally self-dual linear codes over F<SUB>4</SUB>of even lengths, and Choie and Sole (2008) suggested classifying formally self-dual linear codes over F<SUB>4</SUB>of odd lengths in order to study lattices from these codes. These motivate our study on formally self-dual additive codes over F<SUB>4</SUB> In this paper, we define extremal and near-extremal formally self-dual additive codes over F<SUB>4</SUB> classify all extremal codes, and construct many near-extremal codes. We discuss a general method (called the weak balance principle) for constructing such codes. We conclude with some open problems.

Self-dual codes with an automorphism of order 7 and <i>s</i>-extremal codes of length 68

Yankov, Nikolay,Ivanova, Milena,Lee, Moon Ho Elsevier 2018 Finite fields and their applications Vol.51 No.-

<P><B>Abstract</B></P> <P>This paper studies and classifies optimal binary self-dual codes having an automorphism of order 7 with 9 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order <I>p</I>. There are exactly 69781 inequivalent binary self-dual [ 64 , 32 , 12 ] codes with an automorphism of type 7 − ( 9 , 1 ) . As for binary [ 66 , 33 , 12 ] self-dual codes with an automorphism of type 7 − ( 9 , 3 ) there are 1652432 such codes. We also construct more than 4 million new optimal codes of length 68 among which are the first known examples of the very elusive <I>s</I>-extremal self-dual codes. We prove the nonexistence of [ 70 , 35 , 14 ] codes with an automorphism of type 7 − ( 9 , 7 ) . Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.</P>

CYCLIC AND CONSTACYCLIC SELF-DUAL CODES OVER R_k

Karadeniz, Suat,Kelebek, Ismail Gokhan,Yildiz, Bahattin Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4

In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1={\mathbb{F}}_2+u{\mathbb{F}}_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in [1] and we explain why their claim fails.

Cyclic and constacyclic self-dual codes over R_k

Suat Karadeniz,Ismail Gokhan Kelebek,Bahattin Yildiz 대한수학회 2017 대한수학회보 Vol.54 No.4

In this work, we consider constacyclic and cyclic self-dual codes over the rings $R_k$. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over $R_k$ and then construct cyclic self-dual codes over $R_1 = \F_2+u\F_2$ of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-dual codes over $R_1$ of even lengths for which non-trivial such codes exist. In particular we demonstrate that our constructions provide a counter example to a claim made by Batoul et al. in \cite{Batoul} and we explain why their claim fails.

Extremal quasi-cyclic self-dual codes over finite fields

Kim, Hyun Jin,Lee, Yoonjin Elsevier 2018 Finite fields and their applications Vol.52 No.-

<P><B>Abstract</B></P> <P>We study self-dual codes over a factor ring R = <SUB> F q </SUB> [ X ] / ( <SUP> X m </SUP> − 1 ) of length <I>ℓ</I>, equivalently, <I>ℓ</I>-quasi-cyclic self-dual codes of length <I>mℓ</I> over a finite field <SUB> F q </SUB> , provided that the polynomial <SUP> X m </SUP> − 1 has exactly three distinct irreducible factors in <SUB> F q </SUB> [ X ] , where <SUB> F q </SUB> is the finite field of order <I>q</I>. There are two types of the ring R depending on how the conjugation map acts on the minimal ideals of R . We show that every self-dual code over the ring R of the first type with length ≥6 has free rank ≥2. This implies that every <I>ℓ</I>-quasi-cyclic self-dual code of length m ℓ ≥ 6 m over <SUB> F q </SUB> can be obtained by the <I>building-up construction</I>, where <I>m</I> corresponds to the ring R of the first type. On the other hand, there exists a self-dual code of free rank ≤1 over the ring R of the second type. We explicitly determine the forms of generator matrices of all self-dual codes over R of free rank ≤1. For the case that m = 7 , we find 9828 binary 10-quasi-cyclic self-dual codes of length 70 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R of the second type. These codes are all new codes. Furthermore, for the case that m = 17 , we find 1566 binary 4-quasi-cyclic self-dual codes of length 68 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R of the first type.</P>

On some double circulant codes of Crnkovic and disproof of his two conjectures

Heo, D.M.,Kim, J.L. North-Holland Pub. Co 2016 Discrete mathematics Vol.339 No.9

<P>Crnkovic (2014) introduced a self-orthogonal [2q, q - 1] code and a self-dual [2q + 2, q + 1] code over the finite field F-p arising from orbit matrices for Menon designs, for every prime power q, where q 1 (mod 4) and p a prime dividing q+1/2. He showed that if q is a prime and q = 12m +5, where m is a non-negative integer, then the self -dual [2q+2, q + 1] code over F-3 is equivalent to a Pless symmetry code. However for other values of q, he remarked that these codes, up to his knowledge, do not belong to some previously known series of codes. In this paper, we describe an equivalence between his self -dual codes and the known codes introduced by Gaborit in 2002. On the other hand, Cmkovid (2014) also conjectured that if p = 9211 is a prime, the self-orthogonal code and the self -dual code have minimum distance p + 3. We disprove this conjecture by giving two counter-examples in the case of the self-orthogonal code and the self-dual code, respectively when q = 25 and p = 13. (C) 2016 Elsevier B.V. All rights reserved.</P>

이진 극대 자기 쌍대 [14,7,4] 부호를 찾는 빌딩업 방법에 적용된 수정 유전 알고리즘

홍지훈,원병선,김종락 한국지능시스템학회 2024 한국지능시스템학회논문지 Vol.34 No.2

유전 알고리즘(Genetic Algorithm)은 최적 해를 찾는 메타 휴리스틱 알고리즘으로 자연으로부터 유래된 소프트 계산(Soft Computing)의 한 분야이다. 이번 논문에서는 유전 알고리즘을 부호론(Coding Theory) 분야의 이진 극대 자기 쌍대 부호를 찾는 것에 적용해 보았다. 기존의 빌딩업 방법은 모든 짝수 길이의 자기 쌍대 부호를 생성 가능하지만, 최소 거리를 보장할 수 없다. 따라서 유전 알고리즘을 도입하여 큰 최소 거리를 갖는 자기 쌍대 부호를 생성하는 최적화 문제에 도전하였다. 본 논문에서는 유전 알고리즘을 통해 이진 자기 쌍대    부호에서 빌딩업 방법으로 이진 극대 자기 쌍대    부호를 생성하는 최적 해 벡터를 찾는 실험을 소개한다. 본 연구를 통해 극대 자기 쌍대 부호를 찾는 문제를 유전 알고리즘을 활용하여 효과적으로 해결할수 있음을 보였다. The Genetic Algorithm (GA) is a metaheuristic algorithm designed to find optimalsolutions, originating from the field of Soft Computing derived from nature. In thispaper, we aim to apply the Genetic Algorithm to Coding Theory, with a specificfocus on constructing binary extremal self-dual codes. The existing building-up construction technique can generate self-dual codes ofeven length but does not guarantee a minimum distance. Therefore, we haveintroduced the Genetic Algorithm to address the optimization problem of generatingbinary extremal self-dual codes. In this paper, we introduce experiments aimed at finding the optimal solutionvectors for generating binary extremal self-dual    codes from binaryself-dual    codes using the Genetic Algorithm, surpassing the limitations ofthe building-up construction technique in terms of minimum distance. This studydemonstrates the effective resolution of the problem of finding binary extremalself-dual codes using the Genetic Algorithm.

Raising EFL Learners’ Awareness of L2 Lexical Errors and Correct Usage: A Dual Coding Approach

이선정 한국영어교육학회 2017 ENGLISH TEACHING(영어교육) Vol.72 No.2

This study investigated the effects of dual coding elucidation on raising learners’ awareness of L2 lexical errors and correct usage. Participants included 135 Korean EFL middle school students assigned to either a single-coding or dual-coding group. The single-coding group studied the incorrect and correct usage of target lexical items under a verbal-code-only condition. The dual-coding group studied the incorrect and correct usage under a verbal-plus-visual-code condition. Participants completed posttests at two intervals: immediately after studying the materials and four weeks later. Analyses revealed that dual coding elucidation had significant positive effects on facilitating learners’ awareness of lexical errors and correct usage; these effects remained over time. The results also indicated no significant correlations between learning style and the effectiveness of visuals. Qualitative data demonstrated that students perceived visuals as being helpful in improving accurate lexical use and their engagement in learning vocabulary. The article concludes by discussing the facilitative role of visual encoding in L2 lexical knowledge development, thus expanding on the dual coding theory.

On a class of constacyclic codes of length $2p^s$ over $\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}$

Hai Q. Dinh,Bac Trong Nguyen,Songsak Sriboonchitta 대한수학회 2018 대한수학회보 Vol.55 No.4

The aim of this paper is to study the class of $\Lambda$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal R}_a=\frac{\mathbb F_{p^m}[u]}{\left\langle u^a \right\rangle}=\mathbb F_{p^m} + u \mathbb F_{p^m}+ \dots + u^{a-1}\mathbb F_{p^m}$, for all units $\Lambda$ of $\mathcal R_a$ that have the form $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{p^m}$, $\Lambda_0 \,{\not=}\, 0, \, \Lambda_1 \,{\not=}\, 0$. The algebraic structure of all $\Lambda$-constacyclic codes of length $2p^s$ over ${\mathcal R}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.

t-CIS codes over GF(p) and orthogonal arrays

Kim, H.J.,Lee, Y. North Holland ; Elsevier Science Ltd 2017 Discrete applied mathematics Vol.217 No.3

<P>We first show that orthogonal arrays over GF(p) can be explicitly constructed from t-CIS codes over GF(p), where t-CIS codes are CIS codes of order t >= 2. With this motivation, we are interested in developing methods of constructing t-CIS codes over GF(p). We present two types of constructions; the first one is a 't-extension method' which is finding t-CIS codes over GF(p) of length tn from given (t - 1)-CIS codes over GF(p) of length (t - 1)n for t > 2, and the second one is a 'building-up type construction' which is finding t-CIS codes over GF(p) of length t (n + 1) from given t-CIS codes over GF(p) of length tn. Furthermore, we find a criterion for checking equivalence of t-CIS codes over GF(p). We find inequivalent t-CIS codes over GF(p) of length n for t = 3, 4, n = 9, 12, 16, and p = 3, 5, 7 using our construction and criterion, and corresponding orthogonal arrays are found. We point out that 171t-CIS codes we found are optimal codes. (C) 2016 Elsevier B.V. All rights reserved.</P>