Some new results on the minimum length of binary linear codes of dimension nine

“…A printed (but not up to date) table is available in 17 with online and up to date version at. 33 Among others, some of the cases where the problem is solved are n 2 (k, d), k ≤ 8 in, 14 d 3 (n, k), k ≤ 6 (not all cases determined) in, 15 n 4 (5, d) for many cases in, 16 some cases of n 2 (9, d) in, 27 and many cases of n 5 (k, d) for k = 3, 4 in. 13,44 A survey on the subject can be found in.…”

Section: Main Problem Of Coding Theorymentioning

confidence: 99%

Search for Good Linear Codes in the Class of Quasi-Cyclic and Related Codes

Aydın¹,

Asamov²

2010

Selected Topics in Information and Coding Theory

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This chapter gives an introduction to algebraic coding theory and a survey of constructions of some of the well known classes of algebraic block codes such as cyclic codes, BCH codes, Reed-Solomon codes, Hamming codes, quadratic residue codes, and quasi-cyclic (QC) codes. It then describes some recent generalizations of QC codes and open problems related to them. Also discussed in this chapter are elementary bounds on the parameters of a linear code, the main problem of algebraic coding theory, and some algebraic and combinatorial methods of obtaining new codes from existing codes. It also includes a section on codes over Z4, integers modulo 4, due to increased attention given to these codes recently. Moreover, a recently created database of best known codes over Z4 is introduced.

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“…For results on n(9, d) we refer e.g. to [5] and the references therein. The smallest open case for dimension nine is length n = 46 with known bounds 19 ≤ d ≤ 20.…”

Section: Introductionmentioning

confidence: 99%

The $[46, 9, 20]_2$ code is unique

Kurz¹

2021

AMC

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The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case for dimension nine is length n = 46 with known bounds 19 ≤ d ≤ 20. Here we present a [46,9, 20] 2 code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Additionally, we show the non-existence of [47,10, 20] [85,9, 40] 2 codes.

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Some new results on the minimum length of binary linear codes of dimension nine

Cited by 5 publications

References 12 publications

On strongly walk regular graphs, triple sum sets and their codes

On strongly walk regular graphs, triple sum sets and their codes

Search for Good Linear Codes in the Class of Quasi-Cyclic and Related Codes

The $[46, 9, 20]_2$ code is unique

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