Suat Karadeniz

• Ph.D.
• Professor at Texas A&M University – Texarkana

In this work, we consider constacyclic and cyclic self-dual codes over the rings Rk. We start with theoretical existence results for constacyclic and cyclic self-dual codes of any length over Rk and then construct cyclic self-dual codes over R1 = F2 + uF2 of even lengths from lifts of binary cyclic self-dual codes. We classify all free cyclic self-...

In this paper, we give a method to lift binary self-dual codes to the ring . The lifting method requires solving a system of linear equations over . This technique is applied to binary self-dual code to obtain self-dual codes over . As Gray images of these codes, a substantial number of self-dual codes are generated. By using the extension theorem...

In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to . Using this method we construct self-dual codes over of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which hav...

Centraliser codes are codes of length n2n2 defined as centralisers of a given matrix A of order n. Their dimension, parity-check matrices, syndromes, and automorphism groups are investigated. A lower bound on the dimension is n, the order of A. This bound is met when the minimal polynomial is equal to the annihilator, i.e. for so-called cyclic (a.k...

A classification of all four-circulant extremal codes of length 32 over F-2 + uF(2) is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with beta = 80 in W-64,W-2 is f...

In this work, we study construction techniques of formally self-dual codes over the infinite family of rings Rk=F2[u1,u2,…,uk]/〈ui2=0,uiuj=ujui〉. These codes give rise to binary formally self-dual codes. Using these constructions, we obtain a number of good formally self-dual binary codes including even formally self-dual binary codes of parameters...

We give a new construction of the extended binary Golay code. The construction is carried out by taking the Gray image of a self-dual linear code over the ring R = F+2 + uF2 + vF2 + uvF2 of length 6 and size 212.Writing a typical generating matrix of the form [I3|A], with A being a 3×3 matrix over R, and finding some dependencies among the entries...

We consider quasi-cyclic codes over the ring F2 + u F 2 + v F2 + u v F2, a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-...

We obtain extremal binary self-dual codes of parameters [64,32,12] as binary images of self-dual codes over R1, R2 and R3 by employing different methods. We then apply the extension theorem to these codes to obtain a number of extremal binary self-dual codes of length 66 with trivial automorphism groups. Fifteen of the codes we obtain have new β va...

Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and self-dual codes over Z_4+uZ_4 are studied...

A lift of binary self-dual codes to the ring R2 is described. By using this lift, a family of self-dual codes over R2 of length 17 are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length 68 are obtained. For the �rst time in the literature, extremal binary codes of length 68 with = 4 and = 6 in W68;2 ha...

We prove that a putative $[72,36,16]$ code is not the image of linear code over $\ZZ_4$, $\FF_2 + u \FF_2$ or $\FF_2+v\FF_2$, thus proving that the extremal doubly even $[72,36,16]$-binary code cannot have an automorphism group containing a fixed point-free involution. Combining this with the previously proved result by Bouyuklieva that such a code...

In this correspondence, a lift of the shortened binary [8,4,4][8,4,4] Hamming code to the ring R3 is described. Using this lift, a family of self-dual codes over R3 of length 7 are obtained. By taking the binary images of these codes and applying the extension method given in [15], 11 new extremal binary self-dual [58,29,10][58,29,10] codes with ne...

In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructions for self-dual codes over the non-chain ring R 2 = F 2 + uF 2 + vF 2 + uvF 2 are introduced. Using these methods, we have constructed three extremal binary Type I codes of length 64 of new weight enumerators for which extremal codes wer...

Cyclic codes over an infinite family of rings are defined. The general properties of cyclic codes over these rings are studied, in particular nontrivial one-generator cyclic codes are characterized. It is also proved that the binary images of cyclic codes over these rings under the natural Gray map are binary quasi-cyclic codes of index 2k . Furthe...

In this work, (1+v)-constacyclic codes over the ring F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 are studied. (1+v)-constacyclic codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 of odd lengths are characterized with the help of cyclic codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2. A natural Gray map from (F2+uF2+vF2+uvF2)n(F2+uF2+vF2+uvF2)n to (F2+uF2)2n(F2+uF2)2n is intr...

We introduce codes over an infinite family of rings and describe two Gray maps to binary codes which are shown to be equivalent. The Lee weights for the elements of these rings are described and related to the Hamming weights of their binary image. We describe automorphisms in the binary image corresponding to multiplication by units in the ring an...

In this work, we focus on cyclic codes over the ring ${{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}}$ , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring ${({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}...

The focus in this work is on self-dual codes over the ring F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2. Type I and Type II codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 are defined and some of the techniques used in the literature are applied to get some theoretical results about self-dual codes on F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 and their binary images. In particul...

In this work, we investigate linear codes over the ring ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory...