Binary Block Coding 



By S. P. LLOYD 



(Manuscrii)t received March 16, 1956) 



From the work of Shannon one knows that it is possible to signal over an 

 error-making binary channel with arbitrarily small 'probability of error in 

 the delivered information. The effects of errors produced in the channel are 

 to be eliminated, according to Shannon, by using an error correcting code. 

 Shannon's proof that such codes exist does not provide a practical scheme 

 for constructing them, however, and the explicit construction and study of 

 such codes is of considerable interest. 



Particularly simple codes in concept are the ones called here close packed 

 strictly e-error-correcting {the terminology is explained later). It is shown 

 that for such a code to exist, not only 7nust a condition due to Hamming be 

 satisfied, but also another condition. The main result may be put as follows: 

 a close-packed strictly e-error-correcting code on n, n > e, places cannot 

 exist unless e of the coefficient vanish in (1 + xy(l — x)"~^~^ when this is 

 expanded as a polynomial in x. 



I. IXTRODUCTIOX 



In this paper we investigate a certain problem in combinatorial 

 analysis which arises in the theory of error correcting coding. A develop- 

 ment of coding theory is to be found in the papers of Hamming^ and 

 Shannon^; this section is intended primarily as a presentation of the 

 terminology used in subsequent sections. 



We take (0, 1) as the range of binary variables. B}^ an n-word we mean 

 a sequence of n symbols, each of which is or L We call the individual 

 symbols of an 7?-word the letters of the 7?-word. We denote by -B„ the 

 set consisting of all the 2" possible distinct ??-words. The set 5„ may be 

 mapped onto the vertices of an n-dimensional cube, in the usual way, 

 by regarding an n-word as an ?i-dimensional Cartesian coordinate ex- 

 pression. The distance d(u, v) between w-words u and v is defined to be 

 the number of places in which the letters of u and r differ; on the ?z-cube, 

 this is seen to be the smallest number of edges in paths along edges be- 

 tween the vertices corresponding to u and v. The weight of an 7?-word u 



517 



