In an earlier paper the authors studied simplex codes of type α and β over ${\mathbb{Z}}_4$ and o... more In an earlier paper the authors studied simplex codes of type α and β over ${\mathbb{Z}}_4$ and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over ${\mathbb{Z}}_{{2^s}}.$ The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over ${\mathbb{Z}}_{2^{s}}.$ The above codes are also shown to satisfy the chain condition.
This paper studies and classifies linear transformations that connect Hamming distances of codes.... more This paper studies and classifies linear transformations that connect Hamming distances of codes. These include irreducible linear transformations and their concatenations. Their effect on the Hamming weights of codewords is investigated. Both linear and non-linear codes over fields are considered. We construct optimal linear codes and a family of pure binary quantum codes using these transformations.
Self-assembly is a fundamental process by which supramolecular species form spontaneously from th... more Self-assembly is a fundamental process by which supramolecular species form spontaneously from their components. This process is ubiquitous throughout the life chemistry and is central to biological information processing. Algorithms for solving many mathematical and computational problems via tile self assembly has been proposed by many researchers in the last decade. In particular tile set for doing basic arithmetic of two inputs have been given. In this work we give tile set for doing basic arithmetic (addition, subtraction, multiplication) of n inputs and subsequently computing its modulo. We also present a tile set for primality testing. Finally we present a software 'xtilemod' for doing modular arithmetic. This simplifies the task of creating the input files to xgrow simulator for doing basic (addition, subtraction, multiplication and division) as well as modular arithmetic of n inputs. Similar software for creating tile set for primality testing is also given.
This paper gives lower and upper bounds on the covering radius of codes over Z 2 s with respect t... more This paper gives lower and upper bounds on the covering radius of codes over Z 2 s with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type α and Type β) and their dual and give bounds on the covering radii for MacDonald codes of both types over Z 4 .
In an earlier paper the authors studied simplex codes of type α and β over ${\mathbb{Z}}_4$ and o... more In an earlier paper the authors studied simplex codes of type α and β over ${\mathbb{Z}}_4$ and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over ${\mathbb{Z}}_{{2^s}}.$ The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over ${\mathbb{Z}}_{2^{s}}.$ The above codes are also shown to satisfy the chain condition.
This paper studies and classifies linear transformations that connect Hamming distances of codes.... more This paper studies and classifies linear transformations that connect Hamming distances of codes. These include irreducible linear transformations and their concatenations. Their effect on the Hamming weights of codewords is investigated. Both linear and non-linear codes over fields are considered. We construct optimal linear codes and a family of pure binary quantum codes using these transformations.
Self-assembly is a fundamental process by which supramolecular species form spontaneously from th... more Self-assembly is a fundamental process by which supramolecular species form spontaneously from their components. This process is ubiquitous throughout the life chemistry and is central to biological information processing. Algorithms for solving many mathematical and computational problems via tile self assembly has been proposed by many researchers in the last decade. In particular tile set for doing basic arithmetic of two inputs have been given. In this work we give tile set for doing basic arithmetic (addition, subtraction, multiplication) of n inputs and subsequently computing its modulo. We also present a tile set for primality testing. Finally we present a software 'xtilemod' for doing modular arithmetic. This simplifies the task of creating the input files to xgrow simulator for doing basic (addition, subtraction, multiplication and division) as well as modular arithmetic of n inputs. Similar software for creating tile set for primality testing is also given.
This paper gives lower and upper bounds on the covering radius of codes over Z 2 s with respect t... more This paper gives lower and upper bounds on the covering radius of codes over Z 2 s with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type α and Type β) and their dual and give bounds on the covering radii for MacDonald codes of both types over Z 4 .
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Papers by Manish Gupta