Example of encoding process for 5 bit state and 5 packets, N = 2 size bit blocks, m = 3 received packets and f [00] = 01100 , f [01] = 10010 , f [10] = 01010 , f [11] = 10110 transition function, chosen in a pseudorandom way. ⊕ denotes XOR, ” >> ” denotes cyclic shift right by 1 position of the 5 bit state. 

Contexts in source publication

Context 1

... accordingly to the currently encoded N bit block x. There is used a pseudorandomly chosen transition function f : {0, . . . , 2 N − 1} → {0, . . . , 2 64 − 1} for this purpose. The state transition is a cyclic shift of (state XOR f [x]). The encoding procedure is schematically presented as Method 1, example of its application is presented in Fig. 4. Observe that using a pseudorandom number generator (PRNG) initialized with a cryptographic key to choose the f function, we could include encryption in such ...

Context 2

... us start with the Fountain Code (FC) situation. Assume the size of the message is N l and we are receiving m packets of l bits. So among a larger set of packets, we need at least some m ≥ N (undamaged) packets to reconstruct the message. We will now allow the messages to be damaged, as depicted in Fig. 3. For simplicity there will be assumed binary symmetric channel(BSC): packet i ∈ [1 , m ] has i probability of its bits being flipped. Applying forward error correction correspondingly to each channel, the minimal requirement for reconstruction is i (1 − h ( i )) ≥ N . We will get the same bound for JRC. Let us divide the message into length N bit blocks, each block corresponds to a single bit in every packets (vertical lines in Fig. 3). Specifically, the encoding procedure has l steps. In k -th encoding step ( k ∈ { 0 , . . . , l − 1 } ) there are used { N k, ..., N ( k +1) − 1 } bits of the message (bit block x ), to produce k -th bit of every packets. There is required an internal state of encoder to connect redundancy of succeeding blocks. The current implementation uses 64 bit state for this purpose, producing k -th bit of i -th packet as i -th bit of the state . The number of produced packets is limited to 64 this way, the actually received packets correspond to some subset of these 64 bits. The internal state needs to be modified accordingly to currently encoded N bit block x . There is used a pseudorandomly chosen transition function f : { 0 , . . . , 2 N − 1 } → { 0 , . . . , 2 64 − 1 } for this purpose. The state transition is cyclic shift of ( state XOR f [ x ]). The encoding procedure is schematically presented as Algorithm 1, example of its application is presented in Fig. 4. Observe that using a Pseudorandom Number Generator initialized with a cryptographic key to choose the f function, we could include encryption in such encoding. Finally encoding procedure is: set state as some arbitrary initial state (known to receiver), then perform encoding step for blocks 0 to l − 1. The final state: f inal state would be beneficial for decoder for final verification of unidirectional decoding, or is necessary for bidirectional decoding. It can be included in the header of packet. However, unidirectional correction can be performed without it, at cost of probable damage of some last bits of the ...