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Codes in general are often denoted by the letter C, and a code of length n and of rank k (i.e., having n code words in its basis and k rows in its generating matrix) is generally referred to as an (n, k) code. Linear block codes are frequently denoted as [n, k, d] codes, where d refers to the code's minimum Hamming distance between any two code ...
There also exists a Las Vegas construction that takes a random linear code and checks if this code has good Hamming distance, but this construction also has an exponential runtime. For sufficiently large non-prime q and for certain ranges of the variable δ, the Gilbert–Varshamov bound is surpassed by the Tsfasman–Vladut–Zink bound .
Linear block codes are summarized by their symbol alphabets (e.g., binary or ternary) and parameters (n,m,d min) [5] where n is the length of the codeword, in symbols, m is the number of source symbols that will be used for encoding at once, d min is the minimum hamming distance for the code. There are many types of linear block codes, such as
In the mathematics of coding theory, the Griesmer bound, named after James Hugo Griesmer, is a bound on the length of linear binary codes of dimension k and minimum distance d. There is also a very similar version for non-binary codes.
A code has all-symbol locality and availability if every code symbol can be recovered from disjoint repair sets of other symbols, each set of size at most symbols. Such codes are called ( r , t ) a {\displaystyle (r,t)_{a}} -LRC.
These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using Boolean polynomials. Algebraic block codes are typically hard-decoded using algebraic decoders. [jargon]
The Reed–Solomon code is a [n, k, n − k + 1] code; in other words, it is a linear block code of length n (over F) with dimension k and minimum Hamming distance = + The Reed–Solomon code is optimal in the sense that the minimum distance has the maximum value possible for a linear code of size ( n , k ); this is known as the Singleton bound .
Every non-systematic linear code can be transformed into a systematic code with essentially the same properties (i.e., minimum distance). [ 1 ] [ 2 ] Because of the advantages cited above, linear error-correcting codes are therefore generally implemented as systematic codes.