Various constructions for self-dual codes over rings and new binary self-dual codes

Abstract

In this work, extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F 2 m + uF 2 m for m = 1, 2. The duality and distance preserving Gray maps from F 4 + uF 4 to (F 2 + uF 2 ) 2 and F 4 2 are used to obtain self-dual codes whose binary Gray images are [64, 32, 12]-extremal self-dual. An F 2 + uF 2 -extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ = 3 and many codes with the rare γ = 4, 6 parameters are obtained. In addition to these, two hundred fifty doubly even self dual [96, 48,16]-codes with new weight enumerators are obtained from four-circulant codes over F 4 + uF 4 . New extremal doubly even binary codes of lengths 80 and 88 are also found by the F 2 + uF 2 -lifts of binary four circulant codes and a corresponding result about 3-designs is stated.2000 Mathematics Subject Classification. Primary:94B05, Secondary:94B99.

New extremal binary self-dual codes from a Baumert–Hall array

“…The Assmus-Mattson theorem, for example, establishes a strong connection between self-dual codes and designs. This connection was used in [16] to find new 3-designs from Type II extremal binary self-dual codes of length 80.…”

On self-duality and hulls of cyclic codes over $$\frac{\mathbb {F}_{2^m}[u]}{\langle u^k\rangle }$$ with oddly even length

“…There were many literatures on the construction of binary self-dual codes from various kind of codes over F 2 m + uF 2 m for m = 1, 2. For some of these constructions, we refer to [10], [12][13][14][15][16].…”

On s-extremal singly even self-dual [24k+ 8,12k+ 4,4k+ 2] codes

“…Also, we verified by Magma [4] that P 80,1 is equivalent to QR 80 . Very recently, 14 more inequivalent extremal doubly even self-dual codes of length 80 have been found in [15]. These codes are denoted by L 80,i (i = 1, 2, .…”

“…. , 14) in [15,Table 11]. Hence, at least 35 inequivalent extremal doubly even selfdual codes of length 80 exist 1 .…”

“…. , 14) in [15] and the new codes N 80,i (i = 1, 2) as follows. By the Delsarte bound, the covering radius of an extremal doubly even self-dual code of length 80 is at most 14.…”