Two theorems on perfect codes

TWO theorems nre proved on perfect codes. The first cne states tk.at Lloyd's theorem is true without tne assumption that the number of symbols in the alphabet is a prime power. The second thevrem asser?s the impossibility of perfect group codes over non-prime-pcwer-alphabets. IA V be a finite set, I VI = q > 2, and let 1 <_ e 2 ye be Ia:ional integers. We pu?iV= (1, 2,. .. . n). Forv=(Vi)~=l E P, u'= (r#_l E Vn we define d(v, ~1') =: I {i E NI vi + ;:I) I. A perfect e-error-correcting ,code of bloc*k length JZ ovo V is a subset C c Vn suc3 that for every u E I/ " thexe exists exactly oae c E C satisfying d(u, c) <_ e. If 4 is a primie powtir, a necessary condition for the existence of such a code is given by Lloyd's theorem [t;]. This theorem has recently been used to determine all yt, e for which a perfect code over an alphabet V of q symbols, q a prime power, exists [ 5; 63. In $j 1 I show that Lloyd's theorem holds for all q. The proof, which is modelled after [6, 5.41, makes use of some elementary notions from commutative algebra. A different proof has been obtained by I? Del-sarre 4] Z?] , It seems hard to use Lloyd 'b theorem to prove non-existence theorems for perfect codes over non-prime-power-alphabets. ' In $2 I prove the following theorem: if Gi (1 5 i 5 12) is a group with u&,rlying set V, a& C c I$, Gi is a subgroup which as a subset of Vr? is a perfect e-error-correcting code, e < yt, then q is a prime power and each Gi is abeliian of type (p, p, .* ., p). A special case of this theorem was proved in [4].