Low-complexity high-performance low-rate communications codes

1. An encoder apparatus comprising:

an outer repetition encoder for encoding a signal with a mixture of repetition codes with different rates,

an interleaver for permuting encoding from said outer repetition encoder; and

an inner encoder for encoding information from the interleaver to provide a an irregular-repeat zigzag-Hadamard code.

2. The encoder apparatus of claim 1 , wherein a common bit of a zigzag-Hadamard segment of encoding from said inner encoder is a repetition of a last parity bit of a previous zigzag-Hadamard segment of encoding from said inner encoder and said common bit is punctured.

3. The encoder apparatus of claim 1 , wherein a coding rate for said encoder comprise

R

c

=

r

d

v

⁡

(

2

r

-

r

-

1

)

+

r

,

where r is the order of a Hadamard code for said inner encoder and d v are repetitions in resulting codewords from said inner encoder for every information bit.

4. The encoder apparatus of claim 1 , wherein a coding rate for said encoder comprises

R

c

=

r

d

v

⁡

(

2

r

-

r

-

1

)

+

r

,

where r is the order of a Hadamard code for said inner encoder, d v are repetitions in resulting codewords from said inner encoder for every information bit and bits from said information are punctured.

5. The encoder apparatus of claim 1 , wherein said inner encoder comprises un-punctured coding to provide an un-punctured irregular-repeat zigzag-Hadamard code.

6. The encoder apparatus of claim 1 , wherein the inner encoder uses a zig-zag Hadamard code that is based on a two-state Markov process.