Combinatorial S >n-Modules as Codes

Certain -modules related to the kernels ofincidence maps between types in the poset defined by the natural productorder on the set of n-tuples with entries from {1, ,m} are studied as linear codes (whencoefficients are extended to an arbitrary field K). Theirdimensions and minimal weights are computed. The Specht modules areextremal among these submodules. The minimum weight codewords of theSpecht module are shown to be scalar multiples of polytabloids. Ageneralization of t-design arising from the natural permutationS _n-modules labelled by partitions with mparts is introduced. A connection with Reed-Muller codes is noted and acharacteristic free formulation is presented.