Lee and Manhattan MWS and FWS codes
University of New Brunswick
Let q be a prime number and let V = Fnq be the vector space consisting of all the length n vectors whose components are elements of the finite field Fq. We say that C ⊆ Fnq is a linear code if C is a subspace of V , namely for every two elements c1, c2 ∈ C and two scalars a1, a2 ∈ Fq we have a1 · c1 + a2 · c2 ∈ C. The elements of C are called codewords. The finite field Fq which a code is over is called the alphabet. The space F n q may be endowed with a weight function w, which can be induced by a distance metric. Hamming weight, the weight function induced by the Hamming Metric, is the most widely used in coding theory. The size q of the alphabet, and the dimension k of a linear code impose a maximum number of Hamming weights that a linear code can have, denoted LH(k, q). In 2018, Dr. T. Alderson and Dr. A. Neri released a publication titled “Maximum Weight Spectrum Codes” in which it was shown for any prime power q and any positive integer k that LH(k, q) = q k−1 q−1. Linear codes in which there are q k−1 q−1 distinct Hamming weights were given the name maximum weight spectrum (MWS) codes. In this work we determine that for any prime number q and any positive integer k the maximum number of distinct Lee weights that a linear code can have is LL(k, q) = q k−1 2 and the maximum number of distinct Manhattan weights that a linear code can have is LM(k, q) = q k − 1. This is done by first identifying a theoretical upper bound on the functions LL(k, q) and LM(k, q), and then showing the bound is sharp by constructing codes which meet the bound with equality. We show that with respect to Hamming, Lee, and Manhattan weights, there is a lower bound on the length of MWS codes of n = q k−1 q−1. Linear codes in which there is at least one codeword of each weight observed in F n q are called full weight spectrum (FWS) codes. In 2018, Alderson authored an article titled “A note on Full Weight Spectrum Codes” in which it was shown that the maximum length of a Hamming FWS code is n = 2k − 1. We show that with respect to Lee and Manhattan weights, there is an upper bound on the length of FWS codes. Namely, we show that the maximum length of a Lee FWS code is at least n =( q+1 2 ) k−1 ( q+1 2 ) k−1 and the maximum length of a Manhattan FWS code is n = q k−1 q−1. We leave a generalized result for the maximum length of an FWS code with respect to a componentwise metric as an open problem.