Partha Pratim Dey

A Study On Equivalence of 1– Error Correcting Quaternary [5,2] Codes

Abstract In this paper we show that all the 1– error correcting quaternary codes can be partitioned into 4 disjoint equivalence classes.

On symmetric incidence matrices of projective planes

We investigate the incidence matrix of a finite plane of ordern which admits a (C, L)-transitivityG. The elation groupG affords a generalized Hadamard matrixH=(hij) of ordern and an incidence matrix for the plane is completely determined by the matrixR(H)=(R(hij)), whereR(g) denotes the regular permutation matrix forg∈G. We prove that in the caseR(H) is symmetric thatG is an elementary abelian 2-group or elseG is a nonabelian group andn is a square. Results are obtained in the abelian case linking the roots of the incidence matrixR(H) to the roots of the complex matrix χ(H), χ a nontrivial character ofG.

On Quinary [6,4] Error Correcting Codes

Abstract In this paper we investigate the existence, equivalence and weight distribution and some other features of [6, 4] error correcting codes over GF (5).

On complex derivative and complex matrices

Abstract In this paper we explore the square roots of –1 in M 2(R) in order to develop the theory of complex matrices. We are able to show that a root is in fact a two by two matrix of trace zero and determinant one. A complex matrix on the other hand is a multiplication transformation of complex number. In final section we show how complex matrices can be utilized in the study of analyticity of complex functions.

Exploring invariant linear codes through generators and centralizers

Abstract We investigate a H-invariant linear code C over the finite field Fp where H is a group of linear transformations. We show that in the case H is a noncyclic abelian group and (|H|, p) = 1, then the code C is the sum of the centralizer codes Cc (h) where h is a nonidentity element of H. Moreover if A is subgroup of H such that A ≅ Zq × Zq , q ≠ p, then dim C is known when the dimension of Cc (K) is known for each subgroup K ≠ 1 of A. In the last few sections we restrict our scope of investigation to a special class of invariant codes, namely affine codes and their centralizers. New results concerning the dimensions of these codes and their centralizers are obtained.

An investigation of [5,3] error correcting codes over GF(5)

Abstract In papers [1,2], we were able to classify all 1-error correcting [5, 3] and [5, 2] codes over GF(4). In this paper, we investigate the existence and equivalence of [5, 3] error correcting codes over GF(5). It appears that these codes are all mutually equivalent. We also calculate their weight distribution as well as the weight distribution of their orthogonal codes.

Equivalence of error-correcting quaternary [6, 3] self-dual codes

Abstract In an earlier paper [1], we were able to classify all 1-error correcting quaternary [5, 2] codes into 4 disjoint equivalence classes. Pursuing the same ideas, in this paper, we investigate the 1–error correcting self-dual [6, 3] codes over GF(4).We are able to show that these codes are all mutually permutation equivalent.

A study on population dynamics and migration

Abstract In this paper we assume that there are only two communities in a country: majority and minority, and they exhibit an identical proportionality factor, say α. We then proceed to show that if the majority community grows as suggested by Malthusian model but the ratio r (t) of minority to majority population is found to decrease linearly, then that indicates that the members of minority community are disappearing from the country at the exponential rate of α. We are able to prove that this disappearance if continues will eventually lead to the total depletion of minority people from the land in a period of years if r (t) = γ - βt where γ, β > 0. The paper also yields a formula that can be used for projection of total population of the country over the years. In the final section we show some applications of our ideas in the study of demography of Bangladesh.

A note on the code of projective plane of order four

Abstract In this study we discuss a decomposition of p-ary codes C(p, 4) that are obtained from a projective plane of order 4. We are able to show that characteristic vector w ∞ of “line at infinity” belongs to the direct sum Ĉ(p, 4) ⊕ U(p, 4) if and only if p = 2 or p = 5 . Thus for p ≤ 5 , C(p, 4) = Ĉ(p, 4) ⊕ U(p, 4) , otherwise . We use the later decomposition to show that for p > 5 , , whereas we use the former decomposition to produce generator matrices of C(p, 4) for p = 2 or 5.