Communication over Finite-Ring Matrix Channels

Transcription

1 Communication over Finite-Ring Matrix Channels Chen Feng 1 Roberto W. Nóbrega 2 Frank R. Kschischang 1 Danilo Silva 2 1 Department of Electrical and Computer Engineering University of Toronto, Canada 2 Department of Electrical Engineering Federal University of Santa Catarina (UFSC), Brazil IEEE International Symposium on Information Theory Istanbul, Turkey, July 12,

2 Random Linear Network Coding with Errors Transmitter injects packets (row vectors over F q ) Intermediate nodes forward random F q -linear combinations of packets Errors may also be injected, which randomly mix with the legitimate packets (Each) receiver gathers as many packets as possible At any particular receiver: Y = AX + Z where A is a transfer matrix, and Z is some error matrix. 2

3 Finite-Field Matrix Channels A Matrix Channel A Z X + Y Random-linear network-coding with errors can be formulated as: Y = AX + Z, where all matrices are over F q ; X, A, and Z are independent; channel law is specified by the distributions of A and Z. 3

4 Finite-Field Matrix Channels (Cont d) [SKK10] 1 considered three variants of Y = AX + Z over F q. 1 Y = AX : A is invertible, drawn uniformly at random exact capacity, code design, encoding-decoding 2 Y = X + W : W has rank t, drawn uniformly at random exact capacity, code design, encoding-decoding 3 Y = A(X + W ): A invertible, W rank t, both uniform capacity bounds, code design, encoding-decoding 1 Silva, Kschischang, Kötter, Communication over Finite-Field Matrix Channels, IEEE Trans. Inf. Theory, vol. 56, pp , Mar

5 Finite-Ring Matrix Channels Generalize from finite-field matrix channels to finite-ring matrix channels. Why? 2 Nazer and Gastpar, Compute-and-Forward: Harnessing Interference through Structured Codes, IEEE Trans. Inf. Theory, vol. 57, pp , Oct

6 Finite-Ring Matrix Channels Generalize from finite-field matrix channels to finite-ring matrix channels. Why? The motivation comes from physical-layer network coding, in particular, compute-and-forward. 2 2 Nazer and Gastpar, Compute-and-Forward: Harnessing Interference through Structured Codes, IEEE Trans. Inf. Theory, vol. 57, pp , Oct

7 Finite-Ring Matrix Channels: Packet Space uncoded modulation: L 2 -QAM R = Z L [i], packet space = R m, where Z L [i] {a + bi : a, b Z L. nested lattice codes: for many practical constructions, we have 3 : R = T / π tm, packet space = T / π t 1 T / π tm for some t 1 t m, where T is a PID. In all cases, the packet space is R µ for some finite chain ring R, where R µ R {{ R πr {{ πr π s 1 R {{ π s 1 R µ 1 µ 2 µ 1 µ s µ s 1. 3 F., Silva, Kschischang, An Algebraic Approach to Physical-Layer Network Coding, to appear in IEEE Trans. Inf. Theory. 6

8 Finite-Ring Matrix Channels: Packet Space Example: R = Z 4, µ = (3, 5), R µ = Z 3 4 (2Z 4) 2 w = [ ] R µ w = [ ] + 2 [ ] So, the packet space R µ can be visualized as In all cases, the packet space is R µ for some finite chain ring R, where R µ R {{ R πr {{ πr π s 1 R {{ π s 1 R µ 1 µ 2 µ 1 µ s µ s 1. 6

9 Multiplicative Matrix Channel First warmup problem The multiplicative matrix channel (MMC): A X Y Y = AX where X, Y R n µ ; A: invertible, uniform; A and X are independent. 7

10 MMC: Review of [SKK10] When R reduces to F q and R n µ reduces to F n m q : 1 Exact capacity: A preserves the row span, so C MMC = log q ( # of subspaces of F m q ) 2 Capacity-achieving code: reduced row echelon form (RREF) 3 Efficient encoding-decoding: encoding: X = I data n n m n decoding: Gaussian elimination (reduction to RREF) 8

11 MMC: Exact Capacity Theorem The capacity of the MMC, in q-ary symbols per channel use, is C MMC = log q (# of submodules of R µ ). # of submodules of R µ is [ µ ] λ n,µ λ q (see, e.g., [HL00]3 ), where [ µ λ ] q = s i=1 [ ] q (µ i λ i )λ µi i 1 λ i 1, λ i λ i 1 q and [ ] m k is the Gaussian coefficient. q note: λ n, µ means i, λ i n, µ i 3 Honold and Landjev, Linear Codes over Finite Chain Rings, The Electronic J. of Combinatorics, vol. 7,

12 MMC: Capacity-Achieving Code Design code design problem an appropriate generalization of RREF The presence of zero divisors complicates the matters... Over a field, two matrices in echelon form with the same row span will have the same number of nonzero rows the rank. Over a chain ring, this is not the case. For example, the matrices and over Z have the same row span but not the same number of nonzero rows. So, generalization of RREF seems non-trivial. 10

13 MMC: Capacity-Achieving Code Design code design problem an appropriate generalization of RREF There are two matrix canonical forms that generalize RREF: Fuller, A canonical set for matrices over a principal ideal ring modulo m, Canad. J. Math, 54 59, Howell, Spans in the module Z s m, Linear and Multilinear Algebra, 19:1, 67 77,

14 MMC: Capacity-Achieving Code Design code design problem an appropriate generalization of RREF There are two matrix canonical forms that generalize RREF: Fuller, A canonical set for matrices over a principal ideal ring modulo m, Canad. J. Math, 54 59, Howell, Spans in the module Z s m, Linear and Multilinear Algebra, 19:1, 67 77, Example: the matrices and over Z are Fuller and Howell canonical forms, respectively. For details, see our paper and/or Kiermaier s thesis (in German). 10

15 MMC: Efficient Encoding-Decoding First attempt: Encoding: transmit a row canonical form (RCF) Decoding: reduction to RCF The decoding complexity is O(n 2 m), but the encoding is hard. Solution: Encoding: transmit a principal RCF Decoding: reduction to RCF The encoding complexity is O(nm). Principal RCFs occupy a significant portion of all RCFs. Hence, The simple coding scheme asymptotically achieves the capacity. 11

16 Additive Matrix Channel Second warmup problem The additive matrix channel (AMC): W X + Y Y = X + W where X, Y R n µ ; W : shape τ, uniform; W and X are independent. 12

17 Shape of a Matrix The shape is a tuple of non-decreasing integers. Example: µ = (3, 5) R µ = R {{ R πr {{ πr π s 1 R {{ π s 1 R µ 1 µ 2 µ 1 µ s µ s 1. 13

18 Shape of a Matrix The shape is a tuple of non-decreasing integers. Example: µ = (3, 5) R µ = R {{ R πr {{ πr π s 1 R {{ π s 1 R µ 1 µ 2 µ 1 µ s µ s 1. The shape of a module generalizes the concept of dimension. Theorem For any finite R-module M, there is a unique µ such that M = R µ. We call µ the shape of M, and write µ = shape M. 13

19 Shape of a Matrix The shape is a tuple of non-decreasing integers. Example: µ = (3, 5) R µ = R {{ R πr {{ πr π s 1 R {{ π s 1 R µ 1 µ 2 µ 1 µ s µ s 1. The shape of a module generalizes the concept of dimension. Theorem For any finite R-module M, there is a unique µ such that M = R µ. We call µ the shape of M, and write µ = shape M. The shape of a matrix generalizes the concept of rank. Definition The shape of a matrix A is defined as the shape of the row span of A, i.e., shape A = shape(row(a)). 13

20 AMC: Review of [SKK10] When R reduces to F q and R n µ reduces to Fq n m, shape τ reduces to rank t: 1 Exact capacity: a discrete symmetric channel ( ) C AMC = nm log q # of matrices of rank t in F n m q 2 Capacity-approaching code: v is a parameter 0 0 v X =. 0 data n v v m v 3 Efficient encoding-decoding: encoding: error trapping decoding: matrix completion 14

21 AMC: Exact Capacity The AMC is an example of a discrete symmetric channel. Theorem The capacity of the AMC, in q-ary symbols per channel use, is C AMC = log q R n µ log q T τ (R n µ ). We need to derive new enumeration results: R n µ = q n(µ 1+ +µ s). T τ (R n µ ) = [ µ τ ]q Rn τ τ s 1 i=0 (1 qi n ), where [ µ τ ] q = s i=1 [ ] q (µ i τ i )τ µi i 1 τ i 1. τ i τ i 1 q 15

22 AMC: Capacity-Approaching Code Design code design problem a generalization of error-trapping Solution: layered error-trapping Note that every matrix in R n µ admits a π-adic decomposition. Example: R = Z 8, n = 6, µ = (4, 6, 8), X = X 0 + 2X 1 + 4X 2 X 0 = 0 µ 1 X 1 = 0 µ 2 X 2 = µ 3 16

23 AMC: Capacity-Approaching Code Design code design problem a generalization of error-trapping Solution: layered error-trapping Note that every matrix in R n µ admits a π-adic decomposition. Example: R = Z 8, n = 6, µ = (4, 6, 8), X = X 0 + 2X 1 + 4X 2 X 0 = 0 µ 1 X 1 = 0 µ 2 after error-trapping... X 2 = µ 3 X 0 = 0 µ 1 X 1 = 0 X 2 = µ 2 µ 3 16

24 AMC: Efficient Encoding-Decoding Encoding: layered error-trapping, O(nm) complexity Decoding: multistage matrix completion, O(n 2 m) complexity Example: R = Z 8, X = X 0 + 2X 1 + 4X 2. Note that Y = X + W = X 0 + 2X 1 + 4X 2 + W. 1 Take mod 2: [Y ] 2 = X 0 + [W ] 2. 2 Decode X 0 by completing [W ] 2. 3 Clear X 0 from Y : Y = Y X 0 = 2X 1 + 4X 2 + W. 4 Take mod 4: [Y ] 4 = 2X 1 + [W ] 4. 5 Decode 2X 1 by completing [W ] 4. 6 Clear X 1 from Y : Y = Y 2X 1 = 4X 2 + W. 7 We have Y = 4X 2 + W. 8 Decode 4X 2 by completing W. 17

25 Additive-Multiplicative Matrix Channel Now to the main event: The additive-multiplicative matrix channel (AMMC): W A X + Y Y = A(X + W ) where X, Y R n µ ; A: invertible, uniform; W : shape τ, uniform; A, X and W are independent. Remark: This model is statistically identical to Y = AX + Z. 18

26 AMMC: Upper Bound on Capacity Theorem The capacity of the AMMC, in q-ary symbols per channel use, is upper-bounded by C AMMC + log q ( τs+s s s (µ i ξ i )ξ i + i=1 s i=1 (n µ i )τ i + 2s log q 4 + log q ( n+s s τ ) s 1 logq (1 q i n ), where ξ i = min{n, µ i /2. i=0 In particular, when µ 2n, the upper bound reduces to ) C AMMC s (n τ i )(µ i n) + 2s log q 4 i=1 + log q ( n+s s ) + logq ( τs+s s τ ) s 1 logq (1 q i n ). i=0 19

27 AMMC: Coding Scheme coding scheme = principal RCFs + layered error-trapping However, the combination turns out to be non-trivial. Hence, we focus on the special case when τ = (t,..., t). Encoding: 0 0 v X =. 0 X n v v m v Decoding: upon receiving Y = A(X + W ), the decoder simply computes the RCF of Y, which exposes X with high probability. This simple coding scheme asymptotically achieves the capacity for the special case when τ = (t,..., t) and µ 2n. 20

28 Conclusion studied three variants of finite-ring matrix channels exact capacities and an upper bound capacity-achieving codes efficient encoding-decoding methods refined some linear algebra tools over finite chain rings row canonical form with a new proof for uniqueness construction of principal RCFs new enumeration results open problems: Can we handle Y = A(X + W ) for general shapes? What if A is not invertible? 21

29 Back-up Slides 22

30 Finite Chain Rings in One Slide R = π 0 π π 2. π s 1 {0 = π s 23

31 Finite Chain Rings in One Slide R = π 0 π π 2 Let R be a finite chain ring, where π is the unique maximal ideal, q is the order of the residue field R/ π, s is the number of proper ideals. Notation: (q, s) chain ring.. π s 1 {0 = π s 23

32 Finite Chain Rings in One Slide. R = π 0 π π 2 π s 1 {0 = π s Let R be a finite chain ring, where π is the unique maximal ideal, q is the order of the residue field R/ π, s is the number of proper ideals. Notation: (q, s) chain ring. π-adic decomposition Let R(R, π) be a complete set of residues with respect to π. Then every element r R can be written uniquely as r = r 0 + r 1 π + r 2 π r s 1 π s 1 where r i R(R, π). 23

33 Finite Chain Rings: Element Degree Definition The degree, deg(r), of a nonzero element r R, where r = r 0 + r 1 π + + r s 1 π s 1, is defined as the least index j for which r j 0. by convention, deg(0) = s units have degree zero elements of the same degree are associates a divides b if and only if deg(a) deg(b) 24

34 Row Canonical Form Definition A matrix A is in row canonical form if it satisfies the following conditions. 1 Nonzero rows of A are above any zero rows. 2 The pivot of a row is of the form π l, and is the leftmost entry of the least degree. 3 For every pivot (say π l ), all entries below and in the same column as the pivot are zero, and all entries above and in the same column as the pivot are residues of π l. 4 If A has two pivots of the same degree, the one that occurs earlier is above the one that occurs later. If A has two pivots of different degree, the one with smaller degree is above the one with larger degree. For example, over Z 8, A = is in row canonical form. 25

35 Reduction to Row Canonical Form: Example Reduction is a variant of Gaussian elimination. An example over Z 8 : A = A 1 = A 1 = A 2 = A 3 = which is in row canonical form Row canonical form is not necessarily an echelon form! 26

36 Construction of Principal RCFs Definition A row canonical form in T κ (R n µ ) is called principal if its diagonal entries d 1, d 2,..., d r (r = min{n, m) have the following form: d 1,..., d r =1,..., 1, π,..., π,..., π s 1,..., π s 1, 0,..., 0 {{{{{{{{ κ 1 κ 2 κ 1 κ s κ s 1 r κ s. All principal RCFs in T κ (R n µ ) can be constructed via a π-adic decomposition X = X 0 + πx π s 1 X s 1. Example: s = 3, n = 6, µ = (4, 6, 8), and κ = (2, 3, 4) X 0 = κ X 1 = κ X 2 = κ µ 1 µ 2 µ 3 27