Cross-Error Correcting Integer Codes over Z 2

Transcription

1 Cross-Error Correcting Integer Codes over Z 2 m Anna-Lena Trautmann Department of Electrical & Computer Systems Eng., Monash University Department of Electrical & Electronic Engineering, University of Melbourne October 29th, 2014 ISITA 2014, Melbourne In collaboration with Emanuele Viterbo.

2 Introduction Motivation and Introduction Codes over integers are known objects, useful for e.g. coded modulation or magnetic recording. Discretized version of Euclidean metric is Lee metric. 1 / 21

3 Introduction Motivation and Introduction Codes over integers are known objects, useful for e.g. coded modulation or magnetic recording. Discretized version of Euclidean metric is Lee metric. Our special case: Errors only occur in one coordinate with an upper bound on the absolute error value. Then the decoding region (sphere) of a codeword is a cross. 1 / 21

4 Introduction Motivation and Introduction Codes over integers are known objects, useful for e.g. coded modulation or magnetic recording. Discretized version of Euclidean metric is Lee metric. Our special case: Errors only occur in one coordinate with an upper bound on the absolute error value. Then the decoding region (sphere) of a codeword is a cross Lee sphere Cross sphere 1 / 21

5 Introduction Further motivation: We can get larger code rates for cross spheres than for Lee spheres. Example Over Z 16 C L = {(0, 0), (1, 4), (4, 2), (5, 6)} is an optimal 2-Lee-error correcting code. But we can find a larger cross-error correcting code (with error magnitude 2) C + = {(1, 0), (4, 1), (6, 6), (0, 3), (3, 4)}. 2 / 21

6 Introduction Related Work Integer codes for errors in s coordinates with bounded error value (Vinck, Morita 98) Single cross error correcting integers codes over Z p (Morita, Kamada, Kostadinov, van Wijngaarden 07) Single peak shift correcting codes (Levenshtein-Vinck 93, Tamm 97) Works on Lee metric codes (e.g. Golomb-Welch 70, Etzion-Vardy-Yaakobi 10, Etzion 11) 3 / 21

7 Introduction Related Work Integer codes for errors in s coordinates with bounded error value (Vinck, Morita 98) Single cross error correcting integers codes over Z p (Morita, Kamada, Kostadinov, van Wijngaarden 07) Single peak shift correcting codes (Levenshtein-Vinck 93, Tamm 97) Works on Lee metric codes (e.g. Golomb-Welch 70, Etzion-Vardy-Yaakobi 10, Etzion 11) Open question: General results on and constructions of cross error correcting codes over Z 2 m. 3 / 21

8 Mathematical Framework and Bounds 1 Introduction 2 Mathematical Framework and Bounds 3 Linear Constructions for n = 2, 3 4 Decoding 5 Conclusion

9 Mathematical Framework and Bounds Definition (Cross Distance) For any v, w Z n q v i w i if v i w i and v j = w j j i d + (v, w) := 0 if v = w. if i, j : i j, v i w i, v j w j 4 / 21

10 Mathematical Framework and Bounds Definition (Cross Distance) For any v, w Z n q v i w i if v i w i and v j = w j j i d + (v, w) := 0 if v = w. if i, j : i j, v i w i, v j w j Theorem The cross distance sphere with center c and radius t, S + t (c) := {v Zn q d + (v, c) t} is exactly the set of c plus all possible cross errors of magnitude at most t, i.e. S + t (c) = {c + αe i α t}. 4 / 21

11 Mathematical Framework and Bounds Definition (Cross Distance) For any v, w Z n q v i w i if v i w i and v j = w j j i d + (v, w) := 0 if v = w. if i, j : i j, v i w i, v j w j Theorem The cross distance sphere with center c and radius t, S + t (c) := {v Zn q d + (v, c) t} is exactly the set of c plus all possible cross errors of magnitude at most t, i.e. S + t (c) = {c + αe i α t}. = An integer code C is cross-error correcting with error magnitude t iff d + (C) 2t / 21

12 Mathematical Framework and Bounds Lemma A cross sphere in Z n q with radius t and any center c Z n q has volume (c) = 2nt + 1. S + t 5 / 21

13 Mathematical Framework and Bounds Lemma A cross sphere in Z n q with radius t and any center c Z n q has volume (c) = 2nt + 1. Theorem S + t Sphere Packing Bound for C + Z n 2m for error magnitude t: C + Zn 2 m 2nm S t + = (0) 2nt / 21

14 Mathematical Framework and Bounds Lemma A cross sphere in Z n q with radius t and any center c Z n q has volume (c) = 2nt + 1. Theorem S + t Sphere Packing Bound for C + Z n 2m for error magnitude t: C + Zn 2 m 2nm S t + = (0) 2nt + 1. A linear code in Z n 2 m has cardinality 2l, l N. Thus, sphere packing bound for linear codes is the greatest power of 2 below or equal to the above bound. 5 / 21

15 Mathematical Framework and Bounds 2 m C L C + Clin L C + lin Table : Sphere packing bounds on the cardinality of the different codes in Z 2 2m for t = 3. 2 m C L C + Clin L C + lin Table : Sphere packing bounds on the cardinality of the different codes in Z 3 2m for t = 2. 6 / 21

16 Linear Constructions for n = 2, 3 1 Introduction 2 Mathematical Framework and Bounds 3 Linear Constructions for n = 2, 3 4 Decoding 5 Conclusion

17 Linear Constructions for n = 2, 3 Theorem Let k := max{i N 2 i t} and m k + 2. The following is a parity check matrix of a cross code in Z 2 2 m with error magnitude t: ( 2 m k 2 2 H = m k 2 ) 0 2 m k 1. The cardinality of this code is C = 2 2(m k) 3. 7 / 21

18 Linear Constructions for n = 2, 3 Examples: Error magnitude t = 3, over Z 16 : ( ) 2 2 H = 0 4 defines a code of cardinality 8 with generator matrix ( ) 4 4 G =. 0 8 Error magnitude t = 7, over Z 16 : ( ) 1 1 H = 0 2 defines a code of cardinality 2 with generator matrix G = ( 8 8 ). 8 / 21

19 Linear Constructions for n = 2, 3 Error magnitude t = 3, over Z 16 : / 21

20 Linear Constructions for n = 2, 3 Error magnitude t = 3, over Z 64 : / 21

21 Linear Constructions for n = 2, 3 Theorem Let m 4 and t = 2 or t = 3. The code in Z 2 2 m or parity check matrix G = ( 1 t ) generated by H = ( (t + 1) 2 m 4 2 m 4) is a cross code with magnitude t and cardinality 2 2(m t) if m 6. If m = 4 the cardinality is 16 for both t = 2, 3 and if m = 5 the cardinality is 32 for t = 3 and 64 for t = 2. (Typo in paper for m = 5.) 11 / 21

22 Linear Constructions for n = 2, 3 Error magnitude t = 2, over Z 32 : / 21

23 Linear Constructions for n = 2, 3 Error magnitude t = 3, over Z 32 : / 21

24 Linear Constructions for n = 2, 3 Theorem Let k N be maximal such that 2 k t. A parity check matrix of the form ( 2 m k 2 2 H = m k 2 2 m k 2 ) 0 2 m k 1 (2t + 1) 2 m k 2, defines a cross error correcting code in Z 3 2m of magnitude t. 14 / 21

25 Linear Constructions for n = 2, 3 Examples: Error magnitude t = 3, over Z 16 : ( ) H = defines a code of cardinality 64 with generator matrix ( ) G = Error magnitude t = 7, over Z 16 : ( ) H = defines a code of cardinality 16 with generator matrix ( ) G = / 21

26 Decoding 1 Introduction 2 Mathematical Framework and Bounds 3 Linear Constructions for n = 2, 3 4 Decoding 5 Conclusion

27 Decoding Lemma Assume that the error vector e Z n 2 m has only one non-zero coordinate i (i.e. Hamming weight 1) whose value α is in ±{1,..., t}. Then the syndrome vector s = rh T = (c + e)h T = eh T is the α-multiple of (the transpose of) the i-th column of H. 16 / 21

28 Decoding Lemma Assume that the error vector e Z n 2 m has only one non-zero coordinate i (i.e. Hamming weight 1) whose value α is in ±{1,..., t}. Then the syndrome vector s = rh T = (c + e)h T = eh T is the α-multiple of (the transpose of) the i-th column of H. = If we can easily identify the multiples of the columns of H, we can easily syndrome decode our codes. 16 / 21

29 Decoding Decoding algorithm for code with parity check matrix ( 2 m k 2 2 H = m k 2 ) 0 2 m k 1 : Compute the syndromes (s 1 s 2 ) = rh T. IF s 2 = 0 and 2 m k 2 s 1 THEN e := (s 1 /2 m k 2 0). ELSEIF 2s 1 = s 2 and 2 m k 2 s 1 THEN e := (0 s 1 /2 m k 2 ). ELSE return failure. Return c = r e. 17 / 21

30 Decoding Decoding algorithm for code with parity check matrix ( 2 m k 2 2 H = m k 2 ) 0 2 m k 1 : Compute the syndromes (s 1 s 2 ) = rh T. IF s 2 = 0 and 2 m k 2 s 1 THEN e := (s 1 /2 m k 2 0). ELSEIF 2s 1 = s 2 and 2 m k 2 s 1 THEN e := (0 s 1 /2 m k 2 ). ELSE return failure. Return c = r e. 17 / 21

31 Decoding Decoding algorithm for code with parity check matrix ( 2 m k 2 2 H = m k 2 ) 0 2 m k 1 : Compute the syndromes (s 1 s 2 ) = rh T. IF s 2 = 0 and 2 m k 2 s 1 THEN e := (s 1 /2 m k 2 0). ELSEIF 2s 1 = s 2 and 2 m k 2 s 1 THEN e := (0 s 1 /2 m k 2 ). ELSE return failure. Return c = r e. 17 / 21

32 Decoding Decoding algorithm for code with parity check matrix ( 2 m k 2 2 H = m k 2 ) 0 2 m k 1 : Compute the syndromes (s 1 s 2 ) = rh T. IF s 2 = 0 and 2 m k 2 s 1 THEN e := (s 1 /2 m k 2 0). ELSEIF 2s 1 = s 2 and 2 m k 2 s 1 THEN e := (0 s 1 /2 m k 2 ). ELSE return failure. Return c = r e. 17 / 21

33 Decoding Example: Consider the code over Z 16 with t = 3 and ( ) 2 2 H = 0 4 and a received word r = (12 6). Then (s 1 s 2 ) = rh T = (4 8), i.e. 2s 1 = s 2 which means that the error is of the form e = (0 s 1 /2) = (0 2). Hence, we decode to the codeword c = r e = (12 4). 18 / 21

34 Conclusion 1 Introduction 2 Mathematical Framework and Bounds 3 Linear Constructions for n = 2, 3 4 Decoding 5 Conclusion

35 Conclusion Conclusion: We presented constructions for linear cross error correcting codes in Z n 2m for general m and t, for n = 2, 3. These constructions are asymptotically optimal, i.e. achieve the sphere packing bound for m. We presented a decoding algorithm for one of these constructions. Similar algorithms exist for the other constructions. 19 / 21

36 Conclusion Conclusion: We presented constructions for linear cross error correcting codes in Z n 2m for general m and t, for n = 2, 3. These constructions are asymptotically optimal, i.e. achieve the sphere packing bound for m. We presented a decoding algorithm for one of these constructions. Similar algorithms exist for the other constructions. Open question 1: Are some of these codes optimal? Open question 2: For which parameters can we get better cross codes than optimal Lee codes? Open question 3: Constructions for larger n? 19 / 21

37 Conclusion Open question 4: Unions of (shifts of) linear codes?! / 21

38 Conclusion Almost optimal (by factor 2) linear code with t = 2: / 21

39 Thank you for your attention!