Classification of Cyclic Steiner Quadruple Systems

Chang, Yanxun; Fan, Bingli; Feng, Tao; Holt, Derek F.; Östergård, Patric R. J.

doi:10.1002/jcd.21530

Cited by 4 publications

(3 citation statements)

References 29 publications

(98 reference statements)

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“…• a [10,6,4] In spite of Theorem 14, [2q + k, k, 2q] extremal NMDS codes over GF(q) with k ≤ q may exist. It will be shown that extremal NMDS codes yield t-designs for some t. Thus, we are very much fond of extremal NMDS codes.…”

Section: Almost Mds Codes and Near Mds Codesmentioning

confidence: 99%

“…It was shown that the total number of nonisomorphic cyclic Steiner quadruple systems S (3,4,28) is 1028387 [4], which is a big number. This number indicates that it is a hard problem to classify Steiner quadruple systems.…”

Section: Infinite Families Of Near Mds Codes Holding Infinite Familiementioning

confidence: 99%

“…When s = 2, (C (2 s ,2 s +1,3,1) | GF(2) ) ⊥ has parameters [5,4,2], and is MDS. When s = 4, the code has parameters [17,8,6], and is distance-optimal [9].…”

Section: Subfield Subcodes Of the Two Families Of Near Mds Codesmentioning

confidence: 99%

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Infinite Families of Near MDS Codes Holding t-Designs

Ding

Tang

2020

IEEE Trans. Inform. Theory

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An [n, k, n − k + 1] linear code is called an MDS code. An [n, k, n − k] linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system S(5, 6, 12) with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and many infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for t ≥ 2 remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over GF(3 s ) holding an infinite family of 3-designs and an infinite family of near MDS codes over GF(2 2s ) holding an infinite family of 2-designs. The subfield subcodes of these two families are also studied, and are shown to be dimension-optimal or distance-optimal.

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Section: Almost Mds Codes and Near Mds Codesmentioning

confidence: 99%