Classification of extremal double-circulant self-dual codes of length up to 62

IEEE Trans. Inform. Theory

2002

of corresponding to s = s1. It is straightforward to check, using (32), that P ud (C 4 ; ) (2 4 0 1)=2 8 for this . Therefore, C 4 is good. Substituting s = 0:6 in (34) shows that C 4 is improper.We complete the proof of Theorem 1. First, notice from Lemma 18 that for any (n; M ) q code C, C 4 is bad if n = M = q = 2 is not true, and C 5 is bad if n = M = q = 2. It is easy to check that if C is a (2; 2) 2 code of distance one, then C 2 is bad, and therefore, byProposition 1, C m is bad and improper for m 2. On the other hand, Lemma 19 shows that if C is a (2; 2)2 code of distance two, then C 3 is proper and C 4 is good but improper. Such codes are the only codes that are good if used four times. This confirms the results of Theorem 1 in case q = 2.Next, we assume that New Extremal Self-Dual Codes of Length and Related Extremal Self-Dual Codes Radinka Dontcheva and Masaaki Harada

“…These codes are equivalent to the code P60 defined in [7]. Furthermore, two codes with = 0 and with = 10, respectively, were constructed in [13], and a code with = 7 in [4].…”

Section: A Extremal Self-dual Codes Of Length 60mentioning

confidence: 99%

New extremal self-dual codes of length 62 and related extremal self-dual codes

Dontcheva

IEEE Trans. Inform. Theory

2002

“…Using the method given in [9], we have found all extremal self-dual neighbors of the four extremal codes in [8] and two double circulant codes P 50;1 and P 50;2 in [7]. The result is listed in Table II.…”

Section: A New Quasi-symmetric 2-designmentioning

confidence: 99%

A quasi‐symmetric 2‐(49,9,6) design

J of Combinatorial Designs

Munemasa

2002

Self Cite

Huffman and Tonchev discovered four non-isomorphic quasi-symmetric 2-(49,9,6) designs. They arise from extremal self-dual [50,25,10] codes with a certain weight enumerator. In this note, a new quasi-symmetric 2-(49,9,6) design is constructed. This is established by finding a new extremal self-dual [50,25,10] code as a neighbor of one of the four extremal codes discovered by Huffman and Tonchev. A number of new extremal self-dual [50,25,10] codes with other weight enumerators are also found.

“…Thus there cannot exist an extremal formally self-dual even code for these lengths. For self-dual codes, Ward [15] proved that there exist extremal singly-even self-dual codes for lengths 2,4,6,8,12,14,22, 24 and for no other length. Recently, Conway and Sloane [3] obtained an improved upper bound for the minimum weight of singly-even self-dual codes by considering the weight enumerators of the shadows.…”

Section: Theorem 1 (Gleason-piercementioning

confidence: 99%

“…This construction method may be used to construct an extremal singly-even [62,31,12] code from known codes of length 60. Some inequivalent extremal singlyeven [60,30,12] codes were found in [3,7,8].…”

Section: Construction Of a Singly-even [70 35 12] Codementioning

confidence: 99%

“…Ⅲ A few infinite families of formally self-dual even codes were given in [5]. Any even code with generator matrix of the form [I n , B], where B is an n by n symmetric matrix and pure and bordered double circulant even codes (see, e.g., [3,5,8,9] for the constructions) are formally self-dual even codes. It is easy to see that the generator matrices of these codes satisfy the assumptions of Proposition 3.…”

Section: Extremal Formally Self-dual Even Codesmentioning

confidence: 99%

See 1 more Smart Citation

The Existence of a Self-Dual [70, 35, 12] Code and Formally Self-Dual Codes

Finite Fields and Their Applications

1997

Self Cite

In this note, the existence of self-dual codes and formally self-dual even codes is investigated. A construction for self-dual codes is presented, based on extending generator matrices. Using this method, a singly-even self-dual [70, 35,12] code is constructed from a self-dual code of length 68. This is the first published example of a singly-even [70, 35,12] code. Constructions are also given for formally selfdual even codes. Extremal formally self-dual even codes are constructed.