Codes that attain the Hamming bound are called perfect codes. Examples include codes that have only one codeword, and codes that are the whole of A q n {\displaystyle \scriptstyle …

Asymptotic GV Bound for Linear Codes Theorem(AsymptoticGVBound) For every prime power q, p 2(0;1) and "2(0;1 h q(p)), there exists n 0 such that for all n > n 0 there exists an [n;k;d] q …

Definition: Perfect codes A binary block code C is perfect if it attains thesphere-packingboundwithequality. Thus C is a partition of the space Fn 2 into Hamming spheres or …

In 1950, Hamming published a paper that would serve as the basis for modern coding theory. He postulated that it was possible to not only detect, but correct errors in bit strings by calculating …

Apr 30, 2025 · Let be an error-correcting code consisting of codeword s,in which each codeword consists of letters taken from an alphabet of length , and every two distinct codeword s differ in …

Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) \(q\)-ary strings. Any perfect linear code is either a repetition code, a Hamming code, or a …

A binary perfect code of length \(n\) is a code where for some \(e\), every \(n\)-bit word differs from exactly one codeword by at most \(e\) bits. This implies different codewords differ by at least …

Theorem 2 (Singleton Bound). A code with input size n, code length m, and distance ∆ must have length m>n+ ∆. Proof. Let c 1,...,c q ∈{0,1}m be the codewords of our code. Consider …

Bibhas Adhikari (Spring 2022-23, IIT Kharagpur) Information and Coding Theory Lecture 14 March 7, 20235/10 Golay codes This is a perfect code constructed by M. J. E. Golay in 1949.

We refer to the elements of C as words, codewords, or vectors. A code over Fq is called a q-ary code. A code is binary if q = 2, ternary if q = 3, etc. The Hamming distance d(x, y) between x = …