*Perfect Codes***are Space-Time codes for the coherent MIMO channel.**

They are algebraic codes, built on non-commutative fields (or

The channel model considered is the following: if M is the number of transmit and receive antennas,

*division algebras*).The channel model considered is the following: if M is the number of transmit and receive antennas,

Y = H X + N

where H ={hij} is the MxM channel matrix with complex fading coefficients and N the MxM complex Gaussian noise matrix.

A square

*n*STBC is called a_{t}× n_{t}*perfect*code if and only if:*•*It is a full rate linear code using

*n*

^{2}

*information symbols.*

_{t}*•*The determinant of the difference of any two distinct codewords is different from 0 which ensures full-rank and in turn full-diversity.

*•*Non-vanishing determinant is the minimum determinant of a perfect code that is lower bounded away from zero by a constant. This constant is the measure of coding gain.( The coding advantage is an approximate measure of the gain over an uncoded system operating with the same diversity advantage).

*•*The energy required to send the linear combination of the information symbols on each layer is similar to the energy used for sending the symbols themselves (we do not increase the transmitted energy in encoding the information symbols).

*•*It induces uniform average transmitted energy per antenna in all

*T*time slots, i.e., all the coded symbols in the code matrix have the same average energy.

Perfect codes only exist in dimension 2, 3, 4, and 6.

- For M=2 antennas, QAM symbols are sent. There are infinitely many of them, but the most famous is the Golden code. The minimum determinant is 1/5.
- For M=3 antennas, HEX symbols are sent. The minimum determinant is 1/49.
- For M=4 antennas, QAM symbols are sent. The minimum determinant is 1/1125.

For M=6 antennas, HEX symbols are sent. The minimum determinant is between 1/(2

^{6}7^{4}) and 1/(2^{6}7^{5}) .
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