Nearly perfect binary codes

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]....

Perfect binary codes: classification and properties

Perfect binary codes: constructions, properties, and enumeration

Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound we o...

An enumeration of 1-perfect binary codes

We enumerate the extended perfect I-error correcting binary codes of length 16 which can be constructed by the concatenation or doubling construction. In the process, we establish some properties of these codes and consider algorithms that effectively establish the nonequivalence of these codes.

Ternary Hamming and binary perfect covering codes

Given 1 1n € Z, the n-cube Q, is the graph with vertex set V : {0, l}' and ed"geset n:VT:orfi, where f' : {(r,w) e V xV: (u _ w)i:1if and only if j: i,(j:0,.'. ,n-l)), for i:0,... ,n-1. The edges of/t aresaidtohave (ortolie along) the directioni,for i 0,.. . ,n-!.Given a graph G, we say that S gVG) is a perfect dom'inating sei (or PDS) of G if every u € V(G) \ S is neighbor of exactly one verte...