The central problem regarding constant-weight codes is the following: what is the maximum number of codewords in a binary constant-weight code with length <math>n</math>, [[Hamming distance]] <math>d</math>, and weight <math>w</math>? This number is called <math>A(n,d,w)</math>.
Apart from some trivial observations, it is generally impossible to compute these numbers in a straightforward way. Upper bounds are given by several important theorems such as the ''[[first Johnson bound|first]] and [[second Johnson bounds''bound]]s,<ref>See pp. 526–527 of F. J. MacWilliams and N. J. A. Sloane (1979). ''The Theory of Error-Correcting Codes''. Amsterdam: North-Holland.</ref> and better upper bounds can sometimes be found in other ways. Lower bounds are most often found by exhibiting specific codes, either with use of a variety of methods from discrete mathematics, or through heavy computer searching. A large table of such record-breaking codes was published in 1990,<ref name="brouwer">A. E. Brouwer, James B. Shearer, N. J. A. Sloane and Warren D. Smith (1990). "A New Table of Constant Weight Codes". ''IEEE Transactions of Information Theory'' '''36'''.</ref> and an extension to longer codes (but only for those values of <math>d</math> and <math>w</math> which are relevant for the GSM application) was published in 2006.<ref name="smith"/>
