Cocyclic butson Hadamard matrices and codes over Zn via the trace map

Over the past couple of years, trace maps over Galois fields and Galois rings have been used very succesfully o construct cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and subsequently to generate simplex codes over Z4, Z2 and ZP and new linear codes over ZP. Here we define a new map, the trace-like map and more generally the weighted map and extend these techniques to construct cocyclic Budson Hadamard matrices of order (nm) for all n and m and linear and non-linear codes over Zn

New linear codes over Zps via the trace map

The trace map has been used very successfully to generate cocyclic complex and Butson Hadamard matrices and simplex codes over Z4 and Z2s. We extend this technique to obtain new linear codes over Zps. It is worth nothing here that these codes are cocyclic but not simplex codes. Further we find that the construction method also gives Butson Hadamard matrices of order psm

Cocyclic simplex codes of type alpha over Z4 and Z2s

Over the past decade, cocycles have been used to construct Hadamard and generalized Hadamard matrices. This, in turn, has led to the construction of codes-self-dual and others. Here we explore these ideas further to construct cocyclic complex and Butson-Hadamard matrices, and subsequently we use the matrices to construct simplex codes of type /spl alpha/ over Z(4) and Z(2/sup s/), respectively

Complex spherical codes with two inner products

A finite set $X$ in a complex sphere is called a complex spherical $2$-code if the number of inner products between two distinct vectors in $X$ is equal to $2$. In this paper, we characterize the tight complex spherical $2$-codes by doubly regular tournaments, or skew Hadamard matrices. We also give certain maximal 2-codes relating to skew-symmetric $D$-optimal designs. To prove them, we show the smallest embedding dimension of a tournament into a complex sphere by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel matrix.Comment: 10 pages, to appear in European Journal of Combinatoric

A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Some interesting quantum codes are constructed to demonstrate their value.Comment: 33 pages including appendi

Error Correcting Codes Associated with Complex Hadamard Matrices

For primes p \u3e 2, the generalized Hadamard matrix H(p,pt) can be expressed as H = xA, where the notation means hij = xaij. It is shown that the row vectors of A represent a p-ary error correcting code. Depending upon the value of t, either linear or nonlinear codes emerge. Code words are equidistant and have minimum Hamming distance d = (p − 1)t. The code can be extended so as to possess N = p2t code words of length pt − 1