518 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



is the number of I's in the sequence u; it is the distance between u and 

 the »-word 00- • -0, all of whose letters are 0.* 



A binary block code of size K on n places is a class of K nonempty dis- 

 joint subsets of Bn where in each of the K sets a single ?i-word is chosen 

 as the code word of the set.f Each such set is the detection region of the 

 code word it contains, and we shall say that any ?i-word which falls in a 

 detection region belongs to the code word of the detection region. The 

 set consisting of those /«-words which do not lie in any detection region 

 we call limbo.t A close packed code is one for which limbo is an empt}' 

 set; i.e., a code in which the detection regions constitute a partition 

 (disjoint covering) of £„ . 



A sphere of radius r centered at 7i-word u is the set [v:d(u, v) ^ r] of 

 n-words v which differ from u in r or fewer places. A binary block code is 

 e-error-correcting if each detection region includes the sphere of radius e 

 centered at the code word of the detection region. We say that a binary 

 block code is strictly e-error-correcting if each detection region is exactly 

 the sphere of radius e centered at the code word of the detection region. 



This paper is devoted to the consideration of close packed strictly 

 e-error-correcting binary block codes. We shall refer to such a code as 

 an e-code, for bre\dt3'. Hamming observes that a necessary condition 

 for the existence of an e-code on n places is that 



1 + n + I n(n - l) + . . . + (^^ (1) 



be a divisor of 2". In this paper we derive an additional necessary condi- 

 tion. Our condition includes as a special case a condition of Golay for 

 the existence of e-codes of group type, and applies to all e-codes, whether 

 or not they are eciuivalent to group codes. § 



* If Bn is regarded as a subset of the real linear vector space consisting of all 

 sequences a = (ai , 02 , • • • , an) of n real numbers, then the "weight" of an n-word 

 is simply the A norm (defined as || a ||i = ^J" I "" 1^' ^^'^ ^"-'^ "distance" is the 

 metric derived from this norm. 



t The term "block code", due to P. Elias. serves to distinguish the codes of 

 fixed length considered here from the codes of unbounded delay introduced by 

 Elias, Reference 3. 



J In a communications S3-stem- using such a code, the transmitter sends onh- 

 code words. If, due to errors in handling binary symbols, the receiver delivers 

 itself of an n-word other than a code word then: (a) if the «-word lies in a detec- 

 tion region, one assumes that the code word of the detection region was intended; 

 (b) if the n-word lies in limbo, one makes a note to the effect that errors have 

 occurred in handling the word but that one is not attempting to guess what they 

 were. 



§ The terms "group alphal)et" (Slepian^), "systematic code" (Hammingi), 

 "symbol code" (Golay^), "check symbol code" (Elias'), "paritj- check code", are 

 roughly synonymous. More precisely, a group code is a parity check code in which 

 all of the parit}' check forms are homogeneous ("even"), so that 00 ■ • • is one 

 of the code words; see Reference 5. 



