Bar product

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In information theory, the bar product of two linear codes C2 ⊆ C1 is defined as

where (a | b) denotes the concatenation of a and b. If the code words in C1 are of length n, then the code words in C1 | C2 are of length 2n.

The bar product is an especially convenient way of expressing the Reed–Muller RM (dr) code in terms of the Reed–Muller codes RM (d − 1, r) and RM (d − 1, r − 1).

The bar product is also referred to as the | u | u+v | construction[1] or (u | u + v) construction.[2]

Properties[edit]

Rank[edit]

The rank of the bar product is the sum of the two ranks:

Proof[edit]

Let be a basis for and let be a basis for . Then the set

is a basis for the bar product .

Hamming weight[edit]

The Hamming weight w of the bar product is the lesser of (a) twice the weight of C1, and (b) the weight of C2:

Proof[edit]

For all ,

which has weight . Equally

for all and has weight . So minimising over we have

Now let and , not both zero. If then:

If then

so

See also[edit]

References[edit]

  1. ^ F.J. MacWilliams; N.J.A. Sloane (1977). The Theory of Error-Correcting Codes. North-Holland. p. 76. ISBN 0-444-85193-3. 
  2. ^ J.H. van Lint (1992). Introduction to Coding Theory. GTM. 86 (2nd ed.). Springer-Verlag. p. 47. ISBN 3-540-54894-7.