Many codes are zero-divisors and indeed ideals in subalgebras of matrix algebras. For example the cyclic codes are formed from zero-divisor circulant matrices; properties of the codes are inherited from properties of the subalgebra, which in this cyclic case is the subalgebra of circulant matrices.

But algebras also contain units as well as zero-divisors. For example the zero-divisors are the singular matrices and the units are the non-singular (invertible) matrices in the full algebra of matrices.

Here it is shown (initially) how codes may be derived from units in subalgebras of matrices. These codes will then inherit properties of elements in the subalgebra. (The codes are submodules of the subalgebra but are not ideals in general.)

Examples of subalgebras which may be used include circulant-by-circulant (of many types and shapes) and circulants combined with reverse circulants. Indeed these are special cases of group algebras which are subalgebras of matrix algebras.

Low Density Parity Check (LDPC) codes and Convolutional Codes are the ones which at present cause great excitement. These have many applications within the communications area, such as to improve the performance of digital radio, digital video broadcasting, mobile phones, satellite and deep space communications, Bluetooth implementations and elsewhere.

Up to now, there have essentially been no algebraic methods for constructing these; methods used have been computer search for convolutional codes, which restricted the length and (free) distance, and random generation for the LDPC codes.

Properties of the code such as being LDPC will follow directly from properties of group ring elements used in its construction.  This allows algebraic construction of LDPC codes in a straight-forward manner. Similarly Convolutional Codes may be constructed algebraically.

Using properties of difference sets in group ring elements, it is then possible to construct LDPC codes with no short cycles, directly and algebraically. LDPC codes with no short cycles are known to perform exceptionally well.

Examples, simulations and comparisons will be given.