Z4-codes and their gray map images as orthogonal arrays and t-designs

PhDThis thesis discusses various connections between codes over rings, in par- ticular linear Z4-codes and their Gray map images as orthogonal arrays and t-designs. It also introduces the connections between VC-dimension of binary codes and the strengths of the codes as orthogonal arrays. The second chapter concerns codes over rings. It is known that if we have a matrix A over a eld F, whose rows form a linear code, such that any t columns of A are linearly independent then A is an orthogonal array of strength t. I shall begin with generalising this theorem to any nite commutative ring R with identity. The case R = Z4 is particularly important, because of the Gray map, an isometry from Zn 4 (with Lee weight) to Z2n 2 (with Hamming weight). I determine further connections that exist between the strength of a linear code C over Z4 as an orthogonal array, the strength of its Gray map image as an orthogonal array and the minimum Hamming and Lee weights of its dual C?. I also nd that the strength of a binary code as an orthogonal array is less than or equal to its strong VC-dimension. The equality holds for linear binary codes. Furthermore, the lower bound is also determined for the strength of the Gray map image of any linear Z4-code. 4 Moreover, I show that if a code over any alphabet is an orthogonal array with a certain constraint then the supports of the codewords of some Hamming weight form a t-design. Furthermore, I prove that if a linear Z2- code satis es the t-mixture condition, then such a code is an orthogonal array of strength t. I then investigate if such connection also exists for non- linear Gray map images of linear Z4-codes, and prove that it does for some values t

Induced subgraphs of zero-divisor graphs

Funding: Peter J. Cameron acknowledges the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no. EP/R014604/1), where he held a Simons Fellowship. For this research, T. Kavaskar was supported by the University Grant Commissions Start-Up Grant, Government of India grant No. F. 30-464/2019 (BSR) dated 27.03. T. Tamizh Chelvam was supported by CSIR Emeritus Scientist Scheme (No. 21 (1123)/20/EMR-II) of Council of Scientific and Industrial Research, Government of India.The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if ab=0. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph , and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the 2-dimensional dot product graph.Publisher PDFPeer reviewe