Systems and methods for communicating using random codewords located within a restricted subset of a multi-dimensional sphere

1. A communication system, comprising:

a plurality of transmitters that each comprise an encoder, wherein the encoder of a given transmitter is configured to:

receive a start of epoch message;

encode message data as symbols using a rateless code comprising a set of codewords characterized in that they are located within a restricted subset of a multi-dimensional sphere; and

receive feedback messages at a predetermined set of potential decoding times;

wherein the transmitter is configured to transmit the symbols until a received feedback message is an end of epoch message; and

a receiver comprising a decoder, where the decoder is configured to:

cause a broadcast transmitter to transmit at least one start of epoch message;

receive observed symbols;

at each of a predetermined set of decode times, determine whether a decoding rule is satisfied; and

when the decoding rule is satisfied:

decode at least one message based upon the received observed symbols based upon the rateless code; and

cause the broadcast transmitter to transmit an end of epoch message.

2. The communication system of claim 1 , wherein the restricted subset of the multi-dimensional sphere comprises a subset of a surface of the multi-dimensional sphere and an internal portion of the multi-dimensional sphere proximate the subset of the surface of the multi-dimensional sphere.

3. The communication system of claim 1 , wherein the restricted subset of the multi-dimensional sphere comprises a subset of a surface of the multi-dimensional sphere.

4. The communication system of claim 1 , wherein the determination of whether the decoding rule is satisfied is based upon the received observed symbols.

5. The communication system of claim 4 , wherein the decoding rule combines a threshold rule based on a total power received and a maximum likelihood decoding rule.

6. The communication system of claim 1 , wherein:

the codewords located within the restricted subset of the multi-dimensional sphere comprise a number of codewords M of blocklength n K ; and

the multi-dimensional sphere is characterized by a radius equal to √{square root over (n K P)},

wherein P is a maximal (per-codeword) power constraint.

7. The communication system of claim 1 , wherein:

the codewords located within the restricted subset of the multi-dimensional sphere comprise a number of codewords M of blocklength nK that are each concatenated from sub-codewords of blocklengths n 1 , n 2 −n 1 , . . . , n K −n K−1 ; and

each of the sub-codewords of blocklengths n 1 , n 2 −n 1 , . . . , n K −n K−1 is characterized in that sub-codewords of blocklengths n i −n i−1 lie on a restricted subset of a multi-dimensional sphere of radius √{square root over ((n i −n i−1 )P)}.

wherein P is a maximal (per-codeword) power constraint.

8. The communication system of claim 7 , wherein the codewords located within the restricted subset of the multi-dimensional sphere are further characterized in that, when k transmitters from the plurality of transmitters are active, codewords of blocklength nk formed from concatenating the sub-codewords of blocklengths n 1 , n 2 −n 1 , . . . , n K −n K−1 are located within a restricted subset of a multi-dimensional sphere √{square root over (n K P)}.

9. The communication system of claim 1 , wherein multiple transmitters in the plurality of transmitters are configured to select the same rateless code to use in the encoding of the message data as the symbols.

10. The communication system of claim 1 , wherein at least two of the plurality of transmitters are configured to select different rateless codes to use in the encoding of the message data as the symbols.

11. The communication system of claim 10 , wherein the receiver is further configured to apply the decoding rules at separate predetermined times with respect to each of a set of different codebooks.

12. The communication system of claim 1 , wherein the broadcast transmitter is configured to send a negative feedback message when the decoding rule is not satisfied at a predetermined decode time.

13. The communication system of claim 1 , wherein the encoder is configured to continue to encode the message data as the symbols using the rateless code in response to receipt of a negative feedback message.

14. The communication system of claim 1 , wherein the receiver is further configured to measure channel characteristics and select the predetermined set of decoding times based upon the measured channel characteristics.

15. A transmitter system, comprising:

a receiver configured to receive a start of epoch message;

an encoder configured to:

encode message data as symbols using a rateless code comprising a set of codewords characterized in that they are located within a restricted subset of a multi-dimensional sphere; and

a modulator configured to transmit the symbols encoded by the encoder;

wherein the receiver is further configured to receive an end of epoch message at one of a predetermined set of times; and

wherein the modulator is configured to transmit the symbols encoded by the encoder until an end of epoch message is received by the receiver.

16. The transmitter system of claim 15 , wherein the encoder is configured to continue to encode message data as symbols using the rateless code in response to receipt of a negative feedback message.

17. The transmitter system of claim 15 , wherein the restricted subset of the multi-dimensional sphere comprises a subset of a surface of the multi-dimensional sphere and an internal portion of the multi-dimensional sphere proximate the subset of the surface of the multi-dimensional sphere.

18. The transmitter system of claim 15 , wherein the restricted subset of the multi-dimensional sphere comprises a subset of a surface of the multi-dimensional sphere.

19. A receiver system, comprising:

a decoder configured to select at least one codebook;

a transmitter configured to transmit at least one start of epoch message;

a receiver configured to receive observed symbols;

wherein the decoder is further configured to:

determine whether a decoding rule is satisfied at each of a predetermined set of decode times; and

when the decoding rule is satisfied:

decode at least one message based upon the received observed symbols based upon a rateless code comprising a set of codewords characterized in that they are located within a restricted subset of a multi-dimensional sphere; and

cause the transmitter to transmit an end of epoch message.

20. The receiver system of claim 19 , wherein the determination of whether the decoding rule is satisfied is based upon the received observed symbols.

21. The receiver system of claim 19 , wherein the decoding rule combines a threshold rule based on a total power received and a maximum likelihood decoding rule.

22. The receiver system of claim 19 , wherein the transmitter is configured to send a negative feedback message when the decoding rule is not satisfied at a predetermined decode time.

23. The receiver system of claim 19 , wherein the restricted subset of the multi-dimensional sphere comprises a subset of a surface of the multi-dimensional sphere and an internal portion of the multi-dimensional sphere proximate the subset of the surface of the multi-dimensional sphere.

24. The receiver system of claim 19 , wherein the restricted subset of the multi-dimensional sphere comprises a subset of a surface of the multi-dimensional sphere.