On Punctured Reed-Solomon Codes at the Transport Layer of Digital Network

Transcription

1 On Punctured Reed-Solomon Codes at the Transport Layer of Digital Network Shen Fei June, 2010

2 Introduction Basic Knowledge Applying Network Coding at Transport Layer Performance of Using PRS Codes

3 Introduction Problem Assume that a message consisting of k packets should be transmitted to a destination using a digital network with unreliable communication links. The goal is to deliver the message with small delay and with high reliability.

4 Network Model TCP/IP model TCP/IP provides end-to-end connectivity specifying how data should be formatted, addressed, transmitted, routed and received at the destination.

5 Network Model TCP/IP model TCP/IP provides end-to-end connectivity specifying how data should be formatted, addressed, transmitted, routed and received at the destination. Transport Layer The Transport Layer s responsibilities include end-to-end message transfer capabilities independent of the underlying network, along with error control, segmentation, flow control, congestion control, and application addressing (port numbers).

6 Idea of PRS Coding at the Transport Layer Common methods for communication over the packet switching datagram networks employ a feedback channel from receiver to sender in order to control the retransmission of erased or erroneous packets.

7 Idea of PRS Coding at the Transport Layer Common methods for communication over the packet switching datagram networks employ a feedback channel from receiver to sender in order to control the retransmission of erased or erroneous packets. For example: Automatic Repeat-reQuest (ARQ) protocols

8 Idea of PRS Coding at the Transport Layer Common methods for communication over the packet switching datagram networks employ a feedback channel from receiver to sender in order to control the retransmission of erased or erroneous packets. For example: Automatic Repeat-reQuest (ARQ) protocols Disadvantage: wasteful, large redundency

9 Idea of PRS Coding at the Transport Layer Common methods for communication over the packet switching datagram networks employ a feedback channel from receiver to sender in order to control the retransmission of erased or erroneous packets. For example: Automatic Repeat-reQuest (ARQ) protocols Disadvantage: wasteful, large redundency Therefore, it is necessary to apply error correcting codes that require no feedback or almost no feedback to the transport layer of data network.

10 Introduction Basic Knowledge Applying Network Coding at Transport Layer Performance of Using PRS Codes

11 Reed-Solomon Codes Let n different elements from GF (Q) form an ordered set of locators L = {α 0,..., α n 1}.

12 Reed-Solomon Codes Let n different elements from GF (Q) form an ordered set of locators L = {α 0,..., α n 1}. Definition of RS Codes An RS code RS(Q, L; n, k) of length n = L, dimension k and minimum distance d over GF (Q) with the set of locators L GF (Q) consists of all n-words:

13 Reed-Solomon Codes Let n different elements from GF (Q) form an ordered set of locators L = {α 0,..., α n 1}. Definition of RS Codes An RS code RS(Q, L; n, k) of length n = L, dimension k and minimum distance d over GF (Q) with the set of locators L GF (Q) consists of all n-words: c = (c 0, c 1,..., c n 1) = (f (α 0),..., f (α n 1)), where f (x) = f 0 + f 1x f k 1 x k 1 are polynomials over GF (Q) with degree at most k 1.

14 Interleaved Reed-Solomon Codes Definition of IRS Codes Consider all l n matrices C over the field GF (q), where every row of C belongs to the RS code RS(q, L; n, k), L GF (q) \ 0, n = L. The set of all these matrices is called an Interleaved Reed-Solomon (IRS) code of order l.

15 Interleaved Reed-Solomon Codes Definition of IRS Codes Consider all l n matrices C over the field GF (q), where every row of C belongs to the RS code RS(q, L; n, k), L GF (q) \ 0, n = L. The set of all these matrices is called an Interleaved Reed-Solomon (IRS) code of order l. In other words, the IRS code IRS(q l, L; n, k) is IRS(q l, L; n, k) C = c (1) c (2). c (l) c(i) RS(q, L; n, k), i = 1, 2,..., l,

16 Punctured Reed-Solomon Codes Definition of PRS Codes Let GF (Q) = GF (q l ) and L GF (q) \ 0. Then RS(Q, L; n, k) is called Punctured Reed-Solomon (PRS) code. It is RS code with n = L q 1.

17 Punctured Reed-Solomon Codes Definition of PRS Codes Let GF (Q) = GF (q l ) and L GF (q) \ 0. Then RS(Q, L; n, k) is called Punctured Reed-Solomon (PRS) code. It is RS code with n = L q 1. Vector Representation of IRS Codes Represent every column of a code matrix C GF (q l n ) of an Interleaved Reed-Solomon (IRS) code as an element of GF (Q) = GF (q l ), then C c, c GF (Q n ). We obtain vector representation of the IRS code.

18 Punctured Reed-Solomon Codes Definition of PRS Codes Let GF (Q) = GF (q l ) and L GF (q) \ 0. Then RS(Q, L; n, k) is called Punctured Reed-Solomon (PRS) code. It is RS code with n = L q 1. Vector Representation of IRS Codes Represent every column of a code matrix C GF (q l n ) of an Interleaved Reed-Solomon (IRS) code as an element of GF (Q) = GF (q l ), then C c, c GF (Q n ). We obtain vector representation of the IRS code. Lemma Given a set of locators L GF (q), a PRS code RS(q l, L; n, k) is equivalent to an IRS code IRS(q l, L; n, k) in vector representation.

19 Decoding of IRS Codes An efficient decoding method (SSB-decoder, based on multi-sequence shift-register synthesis), corrects up to ε max errors in presence of η erasures, where ε max = l (n k η). l + 1

20 Decoding of IRS Codes An efficient decoding method (SSB-decoder, based on multi-sequence shift-register synthesis), corrects up to ε max errors in presence of η erasures, where ε max = l (n k η). l + 1 If there were ε ε max errors in the Q-ary symmetric channel, the SSB-decoder may fail with probability P f (ε), P f (ε) γq (l+1)(εmax ε) 1, where, γ 1.

21 Decoding of IRS Codes The probability P e of a decoding error is much smaller than P f.

22 Decoding of IRS Codes The probability P e of a decoding error is much smaller than P f. The SSB-algorithm has a computational complexity of order operations in the field GF (q). ℵ = O(ln 2 )

23 Packet Delay over Packet Switching Network Packet Switching Packet switching is a network communications method that groups all transmitted data, irrespective of content, type, or structure into suitably-sized blocks, called packets.

24 Packet Delay over Packet Switching Network Packet Switching Packet switching is a network communications method that groups all transmitted data, irrespective of content, type, or structure into suitably-sized blocks, called packets. Datagram Routing Datagram routing is packet switching in which each packet finds its own path through network according to the current information available at the nodes visited.

25 Average Packet Delay Distribution of Packet Delays Assume that the packet delays are independent random variables t i having exponential distribution with average t i (ρ) = t(ρ) = a 1 ρ, where ρ is the network load and a is a constant for the given network.

26 Delay of the ith Packet of a Message Denote the packet delays in the message in increasing order t 1:k t 2:k t k:k, where t i:k denotes the delay of the ith arriving packet in the total k packets.

27 Delay of the ith Packet of a Message Denote the packet delays in the message in increasing order t 1:k t 2:k t k:k, where t i:k denotes the delay of the ith arriving packet in the total k packets. k E[t i:k ] = t(ρ) j 1 = t(ρ)(h k H k i ), j=k i+1

28 Delay of the ith Packet of a Message Harmonic Number H k is the harmonic number for any natural number k = 1, 2,... : H n = n j 1, j=1

29 Delay of the ith Packet of a Message Harmonic Number H k is the harmonic number for any natural number k = 1, 2,... : H n = n j 1, j=1 Approximations of Harmonic Numbers H 0 = 0, and H n = ln n + γ + 1 2n 1 12n n 4 O(n 6 ), H n ln n + γ, n H n H n i ln n i, where γ is the Euler-Mascheroni constant.

30 Introduction Basic Knowledge Applying Network Coding at Transport Layer Performance of Using PRS Codes

31 Model of the Network

32 Model of the Network The goal is to deliver the message with small delay and with high reliability.

33 Sending Uncoded Message Error Probability for Uncoded Message Denote by p the probability to receive a wrong packet (undetected error, due to unreliable links). P uncoded err = 1 (1 p) k.

34 Sending Uncoded Message Error Probability for Uncoded Message Denote by p the probability to receive a wrong packet (undetected error, due to unreliable links). P uncoded err = 1 (1 p) k. Average Message Delay for Uncoded Scheme For the uncoded case we send k packets of the message and wait until all k packets are received. T uncoded = T (ρ, k, k) = t(ρ)h k a (ln k + γ), 1 ρ

35 Reed-Solomon Coded Scheme Assume that every packet consists of m bits, then a packet can be considered as an element of the field GF (2 m ). The Reed-Solomon (RS) coded scheme is suggested by Kabatiansky and Krouk. A bounded-minimum-distance (BMD) decoder is used.

36 Reed-Solomon Coded Scheme The analysis in RS coded scheme shows that the average delay can be decreased and the reliability of the delivered message can be increased using the proposed encoding if the parameters n and are selected in a proper way.

37 Punctured Reed-Solomon Coded Scheme Select RS code in a special way and to decode it using the method of SSB-decoder [Schmidt-Sidorenko-Bossert (SSB)].

38 Punctured Reed-Solomon Coded Scheme Select RS code in a special way and to decode it using the method of SSB-decoder [Schmidt-Sidorenko-Bossert (SSB)]. Gain in additional decrease of average message delay

39 Punctured Reed-Solomon Coded Scheme Select RS code in a special way and to decode it using the method of SSB-decoder [Schmidt-Sidorenko-Bossert (SSB)]. Gain in additional decrease of average message delay Increase reliability without additional decoding complexity

40 Sending Coded Message Decoding radius of PRS codes is ε max = l (n k η) = 2l l + 1 l + 1, where, η = n k 2 is the number of erasures that can be corrected by PRS decoding.

41 Sending Coded Message Decoding radius of PRS codes is ε max = l (n k η) = 2l l + 1 l + 1, where, η = n k 2 is the number of erasures that can be corrected by PRS decoding. Probability P coded err of a message decoding failure should satisfy the requirement P coded err P err where P err is the acceptable error probability of the message.

42 Sending Coded Message The SSB-decoder may fail (detected failure) with probability Pf (ε): 0 ε ; Pf (ε) = P f (ε) < ε ε max; 1 ε > ε max.

43 Sending Coded Message The SSB-decoder may fail (detected failure) with probability Pf (ε): 0 ε ; Pf (ε) = P f (ε) < ε ε max; 1 ε > ε max. Optimization of Thus, P coded err = + k+2 ε=ε max +1 ( ) k + 2 P f (ε) p ε (1 p) k+2 ε ε ( ) k + 2 p ε (1 p) k+2 ε Perr. ε ε max ε= +1 Now the task is to select to be the minimum integer, such that the equation above is satisfied.

44 Sending Coded Message Average Message Delay by Sending PRS-Coded Message Assume that all network users use the same coding method.

45 Sending Coded Message Average Message Delay by Sending PRS-Coded Message Assume that all network users use the same coding method. The new payload will be increased by 1/R.

46 Sending Coded Message Average Message Delay by Sending PRS-Coded Message Assume that all network users use the same coding method. The new payload will be increased by 1/R. Average delay for a packet will be t(ρ/r) = ar R ρ.

47 Sending Coded Message Average Message Delay by Sending PRS-Coded Message Assume that all network users use the same coding method. The new payload will be increased by 1/R. Average delay for a packet will be t(ρ/r) = Restriction for the code rate: R > ρ. ar R ρ.

48 Sending Coded Message Average Message Delay by Sending PRS-Coded Message Assume that all network users use the same coding method. The new payload will be increased by 1/R. Average delay for a packet will be t(ρ/r) = Restriction for the code rate: R > ρ. ar R ρ. Denote δ = /k, then from k + 2 n, it follows R 1/(1 + 2δ). Finally, we get ( T coded ρ ) = min T ρ<r 1/(1+2δ) R, n, k + 2 ar = min ρ<r 1/(1+2δ) R ρ (Hn H n (k+2 )), where n = k/r. Choose the code rate R in a proper way in order to minimize the mean message delay T coded.

49 Approximate Analysis Using H n H n j ln 1 1 j/n, This allows us to approximate T coded (Fine approximation) as follows T coded For real 0 x < 1 holds Then, T coded = min ρ<r 1/(1+2δ) ln min ρ<r 1/(1+2δ) ar R ρ ln 1 1 R 2δR. 1 1 x < x 1 x. ar R(1 + 2δ) R ρ 1 R(1 + 2δ).

50 Approximate Analysis Minimum T coded take place for R = 2ρ 1 + ρ(1 + 2δ). and the average message delay (Coarse approximation) for this sub-optimized R is T codded 4aρ(1 + 2δ) = [1 ρ(1 + 2δ)]. 2

51 Introduction Basic Knowledge Applying Network Coding at Transport Layer Performance of Using PRS Codes

52 Example Parameters m = 4000 bits Packet length k = 80 packets per message p = 0.03 probability to receive a wrong packet P err = requred probability of wrongly delived message ρ = 0.2 the network load (before coding)

53 Example Parameters m = 4000 bits Packet length k = 80 packets per message p = 0.03 probability to receive a wrong packet P err = requred probability of wrongly delived message ρ = 0.2 the network load (before coding) Select the symbol length s of the interleaved RS code to be s = 8. Then the order of interleaving is l = m/s = 500.

54 Example Parameters m = 4000 bits Packet length k = 80 packets per message p = 0.03 probability to receive a wrong packet P err = requred probability of wrongly delived message ρ = 0.2 the network load (before coding) Select the symbol length s of the interleaved RS code to be s = 8. Then the order of interleaving is l = m/s = 500. Table of Results Transmission method Delay T P err Code rate R ρ/r Uncoded 6.2a KK 1.99a < New 1.41a <

55 Gain in Delay versus Network Load ρ T uncoded coded /T new coded coded T /Tnew KK T uncoded coded /T KK ρ

56 Gain in Delay versus k T uncoded coded /T new coded coded T /Tnew KK T uncoded coded /T KK 4 Gain in delay k

57 Gain in Delay versus p 7 6 T uncoded coded /T new coded coded T /Tnew KK T uncoded coded /T KK p

58 Comparison between new and KK Denote new as the optimal for the PRS coded scheme, KK as the optimal for the RS coded scheme.

59 Comparison between new and KK Denote new as the optimal for the PRS coded scheme, KK as the optimal for the RS coded scheme Δ New Δ KK k

60 Comparison between new and KK Denote new as the optimal for the PRS coded scheme, KK as the optimal for the RS coded scheme Δ New Δ KK Δ New Δ KK k p

61 Approximate Results Coarse Approximation of Message Delay T codded = 4aρ(1 + 2δ) [1 ρ(1 + 2δ)] 2.

62 Approximate Results Coarse Approximation of Message Delay T codded = 4aρ(1 + 2δ) [1 ρ(1 + 2δ)] 2.

63 Approximate Results Fine Approximation of Message Delay T coded min ρ<r 1/(1+2δ) ar R ρ ln 1 1 R 2δR.

64 Approximate Results Fine Approximation of Message Delay T coded min ρ<r 1/(1+2δ) ar R ρ ln 1 1 R 2δR.

65 Reference V. Sidorenko, F. Shen, E. Krouk, M. Bossert, Punctured Reed-Solomon codes at the transport layer of digital networks, The Workshop Coding theory days in St. Petersburg, St. Petersburg, Russia, 6-10 Oct. 2008, pp

66 Additional Slides

67 Additional Slides Process of Reed-Solomon Coded Scheme Encode the k packets of the message into n packets using a RS (n, k) code over GF (2 m ).

68 Additional Slides Process of Reed-Solomon Coded Scheme Encode the k packets of the message into n packets using a RS (n, k) code over GF (2 m ). The destination node waits until k + 2 packets are received.

69 Additional Slides Process of Reed-Solomon Coded Scheme Encode the k packets of the message into n packets using a RS (n, k) code over GF (2 m ). The destination node waits until k + 2 packets are received. a bounded-minimum-distance (BMD) decoder for the RS code is used to reconstruct the message, correcting up to erroneous packets.

70 Thank you!