On the cosets of the simplex code (Q1072523)

Weight and distance properties of cosets of the simplex code are investigated. Let \(S_ m\) be the \((2^ m-1\), m, \(2^{m-1})\) simplex code and S its extension. It is shown that the coset of the all-1 vector of the extended code S of type \((2^ m\), m, \(2^{m-1})\) is the only coset of maximum weight \((2^{m-1})\). The same result is shown for the simplex code of type \((2^ m-1\), m, \(2^{m-1})\). Cosets of \(S_ m\) of weight one less than the maximum, \(2^{m-1}\), are studied. It is established that for \(m\geq 5\) there is no \((2d+1\), m, d) code with \(d\leq 2^{m-2}-2\) for which all vectors have weight at most \(2^{m-2}\). Cosets of weight \(2^{m-2}\) of \(S_ m\) are investigated and the cosets of \(S_ 5\) considered in some detail.