Plotkin's bound in codes equipped with the Euclidean weight function

There are three standard weight (distance) functions on a linear code viz. the Hamming weight (distance), the Lee weight (distance) and the Euclidean weight (distance). Plotkin (11) obtained an upper bound on the minimum weight (distance) of a code with respect to the Hamming weight (distance). A.D. Wyner and R.L. Graham (13) proved Plotkin's bound for Lee metric codes which was also conjectured by Lee (10). The first author also obtained another proof of Plotkin's bound with the Lee weight by a different approach (3). In this paper, we obtain Plotkin's bound for codes equipped with the Euclidean weight function. The Euclidean weight is useful in connection with the lattice constructions where the minimum norm of vectors in the lattice is related to the minimum Euclidean weight of the code (2). Using Plotkin's bound, we obtain a bound on the number of parity check digits required to achieve the minimum Euclidean square distance at least [d.sup.2] in a linear code. We also make a comparative study of the bounds for the Euclidean codes obtained in this paper with the corresponding bounds for the Hamming and Lee weight codes. Keywords and Phrases: Plotkin's bound, Linear code, Euclidean weight.
1. Introduction The minimum distance between any pair of code words in a code cannot exceed the average distance between all pairs of different code words. Using this observation, Plotkin [...]