All Nearly Perfect Codes are Known

Let GF(q) be the finite field of q = pa elements, where p is a prime. Let V be the vector space (GF(q))< The minimum distance drain of a set C _C V is defined as the smallest Hamming distance between different vectors of C. The greatest integer ~ 1, a set C _C V is called an e-error-correcting q-ary code, if ] C I, the cardinality of C, is />2 and e = [(dmin 1)/2]. An e-error-correcting code C is called (Goethals and van Tilborg, 1975) a uniformly packed quasi-perfect code with parameters A, /x, if, for every v ~ V at distance e from some code vector, the number of code vectors at distance e 41 from v is a constant A, and if, for every v c V at distance e + 1 or more from every code vector, the number of code vectors at distance e + 1 from v is a constant/z. IrA and/~ have the maximum values (Bassalygo, Zalcev, and Zinov'ev, 1974; Goethals and van Tilborg, 1975),