Minimal zero-sum sequences in Cn⊕CnCn⊕Cn

Minimal zero-sum sequences of maximal length in Cn⊕CnCn⊕Cn are known to have 2n−12n−1 elements, and this paper presents some new results on the structure of such sequences.It is conjectured that every such sequence contains some group element n−1n−1 times, and this will be proved for sequences consisting of only three distinct group elements. We prove, furthermore, that if pp is an odd prime then any minimal zero-sum sequence of length 2p−12p−1 in Cp⊕CpCp⊕Cp consists of at most pp distinct group elements; this is the best possible, as shown by well-known examples. Moreover, some structural properties of minimal zero-sum sequences in Cp⊕CpCp⊕Cp of length 2p−12p−1 with pp distinct elements are established.The key result proving our second theorem can also be interpreted in terms of Hamming codes, as follows: for an odd prime power qq each linear Hamming code C⊂Fqq+1 contains a non-zero word with letters only 0 and 1.