Abdullah Paşa | Searcharxiv

arXiv2015

The zero short Covering Problem for finite rings

In this work, we find the cardinality of minimal zero short covers of An for any finite local ring A, removing the restriction of D(A)^2 = 0 from the previous works in the literature. Using the structure theorem for Artinian rings, we conclude that we have solved the zero short covering problem for all finite rings. We demonstrate our results on R_k, an infinite family of finite commutative rings extensively studied in coding theory, which satisfy D(A)^2 \neq 0 for all k \geq 2.

math.AC↗

arXiv2014

Extension theorems for self-dual codes over rings and new binary self-dual codes

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 F4 +uF4 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 gamma=3 and many codes with the rare gamma= 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.

cs.IT↗