The triply shortened binary Hamming code is optimal

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n = 2 − 3, 2n−m−1, 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n = 2−4, 2, 3) codeD, i.e., a code with parameters of a triply-shortened Hamming code, is...

The poset structures admitting the extended binary Hamming code to be a perfect code

Brualdi et al. introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Hamming code to be a double-error-correcting perfect P-code. Our study is motivated by this example. In this paper we classify all poset structures which admit the extended binary Hamming code to be a double or triple-error-correcting perfect P-code. © 2004 Elsevier B.V....

Shortened Hamming Codes Maximizing Double Error Detection

Given r ≥ 3 and 2r−1 + 1 ≤ n < 2 − 1, an [n, n − r, 3] shortened Hamming code that can detect a maximal number of double errors is constructed. The optimality of the construction is proven.

On the Undetected Error Probability for Shortened Hamming Codes

New [48,16,16] Optimal Linear Binary Block Code

ALinear binary [n, k]-code is a k-dimensional subspaces of the n-dimensional vector space GF (2) over the finite field GF (2) with 2 elements. The 2 codewords of length n are the elements of the subspace, they are written as row vectors. The weight wt(c) of a codeword is defined to be the number of nonzero entries of c and the minimum distance dist(C) of a code C is the minimum of all weights o...