Talk:Dual code

Types of self-dual codes[edit]

Unless I am very much mistaken, self-dual codes of arbitrary characteristic exist (take any symmetric matrix, and prepend an identity matrix). I added the qualification, from Conway and Sloane's text, that every codeword's weight is a multiple of some constant. Wandrer2 (talk) 13:22, 11 June 2009 (UTC)[reply]

Use of inner product[edit]

I am rather unhappy with the definition on this page. There are two notions of duality for codes over finite fields

(i) duality with respect to the standard symmetric dot product x_1 y_1 + ... + x_n y_n

(ii) duality with respect to the standard sesquilinear dot product x_1 y_1^* + ... + x_n y_n^*

where y |-> y^* represents the order two automorphism of the finite field.

Type (ii) is only relevant when the order of the field is a square.

The proposed definition reduces to (i) when q = p and to (ii) when q = p^2. Otherwise the dual proposed here isn't of a standard type and moreover the double dual of a code won't in general be equal to the original code.

I may try to clarify this on the page itself.

RJChapman (talk) 15:37, 23 November 2007 (UTC)[reply]

I agree, there's no warrant for the definition given here in any standard source and it makes double dual come out wrong. I'll change it if there's no further objection. Richard Pinch (talk) 11:07, 11 July 2008 (UTC)[reply]

Dual code vs Dual space?[edit]

Do dual codes have anything to do with dual space? --Culix (talk) 23:22, 12 March 2008 (UTC)[reply]

Yes indeed, given the points made above, the binary vector space is made into its own dual by use of the inner product. Richard Pinch (talk) 11:07, 11 July 2008 (UTC)[reply]