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ON THE MONOTONIC PROPERTY OF THE PROBABILITY OF UNDETECTED ERROR FOR A shortened code

IEEE TRANSACTIONS ON INFORMATION THEORY 1991年第5期37卷 1409-1412页

作者： FUJIWARA, T KASAMI, T FENG, SP Department of Information and Computer Sciences Faculty of Engineering Science Osaka University Toyonaka Osaka Japan

The monotonic property of the probability of undetected error is considered when a shortened code of a linear code over GF(q) is used for error detection on a q-ary symmetric channel. Some conditions are presented under which the probability of undetected error is (or is not) monotonic with respect to the code length or the symbol error probability. It is shown that the probability of undetected error for a maximum distance separable code is monotonic with respect to the codelength. It is also shown that the probability of undetected error of a shortened code of any binary cyclic Hamming code generated by a trinomial is not monotonic with respect to the bit-error rate if the codelength is short.

关键词： shortened code PROBABILITY OF UNDETECTED ERROR

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Link-level evaluations of low rate code extensions of turbo codes using shortened code techniques

Link-level evaluations of low rate code extensions of turbo ...

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66th IEEE Vehicular Technology Conference

作者： Miyazaki, Shunji Ohbuchi, Kazuhisa Fujitsu Labs Ltd YRP R&D Ctr Kanagawa 2390847 Japan

ISBN: (纸本)9781424402632

In this paper we consider methods of applying shortened code techniques to turbo codes [1] in order to construct codes with a lower code rate than the original turbo codes. As long as the difference of the code rates is not so large, the methods can provide some coding gain compared with bit repetition schemes. For turbo codes, we can consider two types of shortened codes that we term the "outer method" and "inner method." For the UTRA turbo code, which has a code rate of 1/3, simulations show that the performance gain is about 0.2 dB for the outer method and about 0.4 dB for the inner method, and the performance is degraded at most by about 0.1 dB even compared with a dedicated turbo code with a rate of 1/5 for rates of between 1/9 and 1/7. The complexities of the decoder for the methods are also considered. The complexities slightly increase compared with the bit repetition schemes based on the same code formats but are similar to the dedicated code.

关键词： turbo code low rate code extension shortened code Max-Log-MAP

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Some 3-designs and shortened codes from binary cyclic codes with three zeros

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FINITE FIELDS AND THEIR APPLICATIONS 2023年第1期89卷

作者： Xiang, Can Tang, Chunming South China Agr Univ Coll Math & Informat Guangzhou 510642 Guangdong Peoples R China Southwest Jiaotong Univ Sch Informat Sci & Technol Chengdu 610031 Peoples R China

Linear codes and t-designs are interactive with each other. It is well known that some t-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of t-designs with t = 3are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold 3-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal. (c) 2023 Elsevier Inc. All rights reserved.

关键词： Linear code Cyclic code shortened code t-designs

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shortened Linear codes From APN and PN Functions

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第6期68卷 3780-3795页

作者： Xiang, Can Tang, Chunming Ding, Cunsheng South China Agr Univ Coll Math & Informat Guangzhou 510642 Guangdong Peoples R China China West Normal Univ Sch Math & Informat Nanchong 637002 Sichuan Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some p-ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and p-ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes.

关键词： Linear code shortened code PN function APN function t-design

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shortened Linear codes Over Finite Fields

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第8期67卷 5119-5132页

作者： Liu, Yang Ding, Cunsheng Tang, Chunming Tianjin Univ Commerce Coll Math & Informat Sci Tianjin 300384 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China China West Normal Univ Sch Math & Informat Nanchong 637002 Peoples R China

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper.

关键词： Linear code cyclic code punctured code shortened code t-design

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Information Retrieval With Varying Number of Input Clues

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IEEE TRANSACTIONS ON INFORMATION THEORY 2016年第2期62卷 625-638页

作者： Junnila, Ville Laihonen, Tero Univ Turku Dept Math & Stat FI-20014 Turku Finland

Information retrieval in associative memories was studied in a recent paper by Yaakobi and Bruck (2012). Associations between memory entries give us the t-neighbourhood of an entry. In their model, an information unit is retrieved from the memory with the aid of input clues, which are chosen from a reference set. In this paper, we consider the situation where the information unit is found unambiguously using the associated t-neighbourhoods of the input clues. A varying number of input clues are allowed, but a limit m(u) on the maximum number of them is imposed. Of course, we would like m(u) to be as small as possible. We consider the problem over the binary Hamming space F-n and focus on the minimum of m(u), denoted by.(n;t). Using linear reference sets, we show that.(n;2) <= 5 for any n >= 9. We also give infinite families of reference sets, which provide good bounds on.(n;t) for t = 3. In addition, efficient methods are given to obtain bounds on.(n;t) for any t from known reference sets. We also discuss the applications of this model to the Levenshtein's sequence reconstruction problem and the sensor network monitoring.

关键词： Information retrieval associative memory Hamming space shortened code linear code

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Some Punctured codes of Several Families of Binary Linear codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第8期67卷 5133-5148页

作者： Wang, Xiaoqiang Zheng, Dabin Ding, Cunsheng Hubei Univ Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f) = {Tr(af (x) + bx) x is an element of F-qm* :a, b is an element of F-q(m)}, where q is a prime power, F-q(m) = F-q(m) \ {0}, Tr is the trace function from F-q(m) to F-q, and f (x) is a function from F-q(m) to F-q(m) with f (0) = 0. Almost bent functions, quadratic functions and some monomials on F-2(m) were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.

关键词： Boolean function linear code punctured code shortened code weight distribution

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Generalized Subspace Subcodes With Application in Cryptology

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IEEE TRANSACTIONS ON INFORMATION THEORY 2019年第8期65卷 4641-4657页

作者： Berger, Thierry P. Gueye, Cheikh Thiecoumba Klamti, Jean Belo Univ Limoges XLIM UMR 7252 F-87000 Limoges France Univ Cheikh Anta Diop Fac Sci & Tech DMI LACGAA Dakar 5005 Senegal

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code. In this paper, we explore in detail the subspace subcodes construction. This kind of codes was already studied in the particular case of cyclic Reed-Solomon codes. We extend this approach to any linear code over the extension of a finite field. We are interested in additive codes who are deeply connected to subfield subcodes. We characterize the duals of subspace subcodes. We introduce the notion of generalized subspace subcodes. We apply our results to generalized Reed-Solomon codes which leads to codes with interesting parameters, especially over a large alphabet. To conclude this paper, we discuss the security of the use of generalized subspace subcodes of Reed-Solomon codes in a cryptographic context.

关键词： Additive code subfield subcode subspace subcode punctured code shortened code projected code

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A Hybrid ARQ Scheme Based on Rate-Compatible Low-Density Parity-Check codes by Shortening and Extending

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2009年第11期E92A卷 2883-2890页

作者： Okamura, Toshihiko NEC Corp Ltd Common Platform Software Res Labs Kawasaki Kanagawa 2118666 Japan

Incremental Redundancy Hybrid ARQ (IR-HARQ) based on rate-compatible punctured low-density parity-check (LDPC) codes call achieve high throughput over a wide range of SNRs. One drawback of such IR-HARQ schemes is high computational complexity of decoding for early transmission at high rates. In order to overcome this problem, a HARQ scheme based on rate-compatible LDPC codes by shortening and extending is presented in this paper. In the HARQ scheme, a high-rate mother code is transmitted at first, and parity-bits of a shortened code are transmitted for early retransmission requests. With a low-complexity decoder of the high-rate mother code, this shortened-code approach would result in low computational complexity of decoding, but it causes smaller length and larger number of shortened codes to be decoded as retransmission repeats. To prevent the resultant degradation of performance and complexity, extending is efficiently applied to the shortened codes after predetermined retransmission-times. A multi-edge type code-design is employed to construct irregular LDPC codes that meet the requirement of the HARQ scheme. Simulation results show that the HARQ scheme can achieve lower computational complexity of decoding than a conventional IR-HARQ scheme with good throughput over a wide range of SNRs.

关键词： low-density parity-check code hybrid ARQ shortened code

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The minimum locality of linear codes

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DESIGNS codeS AND CRYPTOGRAPHY 2023年第1期91卷 83-114页

作者： Tan, Pan Fan, Cuiling Ding, Cunsheng Tang, Chunming Zhou, Zhengchun Yunnan Normal Univ Sch Informat Sci & Technol Kunming 650500 Yunnan Peoples R China Southwest Jiaotong Univ Sch Math Chengdu 611756 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Kowloon Clear Water Bay Hong Kong Peoples R China China West Normal Univ Sch Math & Informat Nanchong 637002 Sichuan Peoples R China

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.

关键词： Cyclic code Linear code Locally recoverable code Near MDS code Punctured code shortened code

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