The MacWilliams identity for mm-spotty weight enumerator over F2+uF2+⋯+um−1F2\mathbb{F}_2+u\mathbb{F}_2+\cdots+u^{m-1}\mathbb{F}_2

Past few years have seen an extensive use of RAM chips with wide I/O data (e.g. 16, 32, 64 bits) in computer memory systems. These chips are highly vulnerable to a special type of byte error, called an $m$-spotty byte error, which can be effectively detected or corrected using byte error-control codes. The MacWilliams identity provides the relationship between the weight distribution of a code and that of its dual. The main purpose of this paper is to present a version of the MacWilliams identity for $m$-spotty weight enumerators over \mathbbm{F}_{2}+u\mathbbm{F}_{2}+\cdots+u^{m-1}\mathbbm{F}_{2} (shortly $R_{u, m, 2}$).Comment: Research paper, under review since 18th October 2012. arXiv admin note: substantial text overlap with arXiv:1307.178

MacWilliams Type identities for mm-spotty Rosenbloom-Tsfasman weight enumerators over finite commutative Frobenius rings

The $m$-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). $m$-spotty byte error control codes are very suitable for burst correction. M. \"{O}zen and V. Siap [7] proved a MacWilliams identity for the $m$-spotty Rosenbloom-Tsfasman (shortly RT) weight enumerators of binary codes. The main purpose of this paper is to present the MacWilliams type identities for $m$-spotty RT weight enumerators of linear codes over finite commutative Frobenius rings.Comment: Research article, orignial manuscript under review since 2nd November 2012. 9 pages, 4 Tables. arXiv admin note: substantial text overlap with arXiv:1307.1786, arXiv:1307.222

MacWilliams type identities for some new mm-spotty weight enumerators over finite commutative Frobenius rings

Past few years have seen an extensive use of RAM chips with wide I/O data (e.g. 16, 32, 64 bits) in computer memory systems. These chips are highly vulnerable to a special type of byte error, called an $m$-spotty byte error, which can be effectively detected or corrected using byte error-control codes. The MacWilliams identity provides the relationship between the weight distribution of a code and that of its dual. This paper introduces $m$-spotty Hamming weight enumerator, joint $m$-spotty Hamming weight enumerator and split $m$-spotty Hamming weight enumerator for byte error-control codes over finite commutative Frobenius rings as well as $m$-spotty Lee weight enumerator over an infinite family of rings. In addition, MacWilliams type identities are also derived for these enumerators.Comment: Research article, under review since 30th March 2013. 18 pages,6 Tables. arXiv admin note: text overlap with arXiv:1109.3800 by other author