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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]
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 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]