In this paper, we will review some extremal doubly even self dual
(type II) codes of length divide by 8, and discuss the constructions of
these codes. A type II [8, 4, 4] code isomorphic to the Hamming code
is constructed by using quadratic residues. Then we use the direct
sum of two [8, 4, 4] codes to obtain a type II [16, 8, 4] code. A type II
[24, 12, 8] code isomorphic to the Golay code will also be constructed.
It can be shown that it is unique by using the theory of designs. We
also construct a type II [32, 16, 8] code by using the Hadamard matrix.
In fact, it can shown that all type II codes of length 8, 16, and 24, can
be constructed by some Hadamard matrices of orders 8, 16, and 24,
respectively. Finally, we will also introduce the notion of Reed-Muller
codes (RM codes) and discuss its properties.

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