Parents, children, neighbors and the shadow [binary code theorems]

Discusses five code theorems. The author brings together several concepts with interesting relations to each other. All the codes are binary. The weights of all vectors in a self-orthogonal code must be even, however all weights in a code can be even without the code being self-orthogonal. The author calls C even if all its weights are even, and calls C doubly-even (d.e.), if the weights of all vectors in C are divisible by 4. A vector whose weight is divisible by 4 is also called d.e. An even code which is not doubly-even is called singly-even (s.e.). The author calls an even code balanced if it contains the same number of vectors whose weights are ≡0(mod 4) as those of weights ≡2(mod 4). Any s.e. self-orthogonal code is balanced. The author calls a coset balanced if either all weights in it are even and half are ≡0(mod 4), half ≡2(mod 4) or all weights are odd and half are ≡1(mod 4), half ≡3(mod 4)