Minimal zero-sum sequences in

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