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Perfect and Related Codes

Perfect and Related Codes. OUTLINE. [1] Some bounds for codes [2] Perfect codes [3] Hamming codes [4] Extended codes [5] The extended Golay code [6] Decoding the extended Golay code [7] The Golay code [8] Reed-Muller codes [9] Fast decoding RM(1,m). Perfect and Related Codes.

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Perfect and Related Codes

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  1. Perfect and Related Codes

  2. OUTLINE • [1] Some bounds for codes • [2] Perfect codes • [3] Hamming codes • [4]Extended codes • [5] The extended Golay code • [6] Decoding the extended Golay code • [7] The Golay code • [8] Reed-Muller codes • [9] Fast decoding RM(1,m)

  3. Perfect and Related Codes • [1] Some bounds for codes • 1. The number of word of length n , weight t • 2. Theorem 3.1.1

  4. Perfect and Related Codes • 3. Theorem 3.1.3 Hamming bound(upper bound) • C: a code of length n, distance d = 2t+1 or 2t+2 • Eg 3.1.4 Give an upper bound of the size of a linear code C of length n=6 and distance d=3 So but the size of a linear code C must be a power of 2 so

  5. Perfect and Related Codes • 4. Theorem 3.1.7 Singleton bound(upper bound) • For any (n, k, d) linear codes, d-1≦ n-k (i.e. k ≦ n-d+1 or |C| ≦2n-d+1 ) <pf> the parity check matrix H of an (n,k,d) linear code is an n by n-k matrix such that every d-1 rows of H are independent. Since the rows have length n-k, we can never have more than n-k independent row vectors. Hence d-1≦ n-k. • 5. Theorem 3.1.8 For a (n, k, d) linear code C, the following are equivalent: • d = n-k+1 • Every n-k rows of parity check matrix are linearly independent • Every k columns of the generator matrix are linearly independent • C is Maximum Distance Separable(MDS) (definition: if d=n-k+1)

  6. Perfect and Related Codes • 6. Theorem 3.1.13 Gilbert-Varshamov condition • There exists a linear code of length n, dimension k and distance d if (<pf> design a parity check matrix under this condition. See Ex3.1.22) • 7. Corollary 3.1.14 Gilbert-Varshamov bound(lower bound) • If n≠1 and d ≠1, there exists a linear code C of length n and distance at least d with (<pf> choose k such that then |C| = 2k = )

  7. Perfect and Related Codes • Eg 3.1.15 Does there exist a linear code of length n=9, dimension k=2, and distance d=5? Yes, because • Eg 3.1.16 What is a lower and an upper bound on the size or the dimension, k, of a linear code with n=9 and d=5? G-V lower bound: |C| ≧ but |C| is a power of 2 so |C| ≧ 4 Hamming upper bound: |C| ≦ but |C| is a power of 2 so |C| ≦ 8

  8. Perfect and Related Codes • Eg 3.1.17Does there exists a (15, 7, 5) linear code? Check G-V condition G-V condition does not hold, so G-V bound does not tell us Whether or not such a code exists. But actually such a code does exist. (See BCH code later)

  9. Perfect and Related Codes • [2]. Perfect Codes • 1. Definition: • A code C of length n and odd distance d = 2t+1 is called perfect code if • 2. Theorem 3.2.8 • If C is perfect code of length code of length n and distance d = 2t+1, then C will correct all error pattern of weight less than or equal t

  10. Perfect and Related Codes • [3]. Hamming Codes • 1. Definition: Hamming code of length 2r-1 • A code of length n = 2r-1, r ≧2, having parity check matrix H whose rows of all nonzero vectors of length r • Eg 3.3.1 the Hamming code of length 7(r=3)

  11. Perfect and Related Codes • H contains r rows of weight one, so its r columns are linearly independent. Thus a Hamming code has dimension k=2r-1-r and contains 2k codewords. • It is a perfect single error-correcting code (d=3) 1. Any two rows of H are lin. indep so d ≧3 2. 100…0, 010…0, and 110…0 are lin. dep. so d ≦3 and thus d=3 3. Attains the Hamming bound (t=1)

  12. Perfect and Related Codes • [4]Extended Codes • 1. Definition: A code C* of length n+1 obtained from C of length n Construction: Let G: kxn and choose G*=[G, b] : kx(n+1) where the last col b of G* is appended so that each row of G* has even weight. Let H: nx(n-k) then H* = where j is the nx1 col of all ones. Why? where Gj+b=0

  13. Perfect and Related Codes • Eg 3.4.1 C: length 5 C*: length 6

  14. Perfect and Related Codes • [5]. The extended Golay code(C24) • 1. Definition the linear code C24 with generator matrix G= [I, B] I: 12x12 identity matrix B:

  15. Perfect and Related Codes • 2. Important facts of the extended Golary Code C24 • Length n =24, dimension k= 12 , 212=4096 codewords • Parity check matrix • Another parity check matrix • Another generator matrix [B, I] • C24 self-dual; • The distance of C is 8 • C24 is a three-error-correcting code

  16. Perfect and Related Codes • [6]. Decoding the extended Golay code • 1. Algorithm 3.6.1 IMLD for Code C24 • w: received word • 1) s = wH • 2) if wt(s)≦3 then u = [ s,0 ](error pattern) • 3) if wt(s+bi) ≦2 for some row bi of B then u=[s+bi, ei] • 4) compute the second syndrome sB • 5) if wt(sB)≦3 then u=[0, sB] • 6) if wt(sB+bi)≦2 for some row bi of B then u=[ei, sB+bi] • 7) if u is not yet determined then request retransmission

  17. Perfect and Related Codes • 2. Eg. 3.6.2 decode w = 101111101111, 010010010010

  18. Perfect and Related Codes • [7]. The Golay code • 1. C23 • Removing a digit from every word in C24 • : 12 x 11 matrix by deleting the last column of B • G: 12x23 matrix, G=[I12, ] • Length n=23, dimension k=12, 212=4096 words, distance d=7 • Perfect code, three-error-correcting code • Algorithm 3.7.1 (decoding) • Form w0 or w1, which has odd weight • Decode wi(I is 0 or 1) using Algorithm 3.6.1 to a codeword c in C24 • Removing the last digit from c

  19. Perfect and Related Codes • [8]. Reed-Muller codes • 1. r-th order, length 2m, 0≦r≦m, RM(r, m) • RM(0, m)={00…0, 11…1}, RM(m, m)= • RM(r, m) = • 2. Eg 3.8.1

  20. Perfect and Related Codes • 3. Generator matrix G of RM(r, m) • 4. Eg 3.8.4 Find G(1,3)

  21. Perfect and Related Codes • 5. The properties of RM(r,m)

  22. Perfect and Related Codes • [9]. Fast decoding for RM(1,m) • 1. The Kronecker product of matrices • A x B = [aijB] • Eg 3.9.1 • Definition:

  23. Perfect and Related Codes • 2. Algorithm 3.9.4 decoding the RM(1,m) code codeword v = presumed message m x G(1,m)

  24. Perfect and Related Codes • Eg 3.9.5 m=3, w=10101011 codeword

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