Reed-Muller Codes. These codes were discovered by Muller and the decoding by Reed in Code length: n = 2 m, Dimension: Minimum Distance

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1 Reed-Muller Codes Ammar Abh-Hhdrohss Islamic University -Gaza ١ Reed-Muller Codes These codes were discovered by Muller and the decoding by Reed in 954. Code length: n = 2 m, Dimension: Minimum Distance For m =5 and r = 2 then n = 32, k = 6, and d min = 8. Slide ٢ ١

2 Reed-Muller Codes For m =5 and r = 2 then n = 32, k = 6, and d min = 8. For i m, let v i be a 2 m -tuple over GF(2) of the following form: For m =4, we have the following four vectors v 4 = (,,,,,,,,,,,,,,,) v 3 = (,,,,,,,,,,,,,,,) v 2 = (,,,,,,,,,,,,,,,) v = (,,,,,,,,,,,,,,,) Slide ٣ Reed-Muller Codes Let a = (a, a,., a n- ) and b = (b, b,., b n- ) be a two binary n- tuples. The Boolean product of a and b is defined as: a.b For example a. b, a. b,,. a n b n The product vector v i v i2 v il is said to have degree l. Slide ٤ ٢

3 ٣ Slide ٥ Reed-Muller Codes Because the weights of v, v 2,., v m are even and powers of 2, their products have weights even and a power of 2. The r th order RM is spanned by the following set of independent vectors We have the following inclusion chain Slide ٦ Example Let m =4 and r = 2, the RM code is generated by the following vectors v v 4 v 3 v 2 v v 3 v 4 v 2 v 4 v v 4 v 2 v 3 v v 3 v v 2

4 Reed Decoding The importance of Muller code stems from the reduced complexity of the decoding process. This process pioneered by Reed is better explained by an example Consider the RM(2, 4) and consider the message to be encoded The corresponding codeword is Slide ٧ The sum of the first four components of each generator vector is zero except for v v 2. The same applied three groups of four consecutive components. Or Now let r = (r, r,, r 5 ) be the received vector. a 2 can be decoded using Slide ٨ ٤

5 a 2 is taken to be the majority of {A, A 2, A 3, A 4 ) If there is only one error a 2 can be detected correctly. The same can be applied for a 3 The same can be applied for a 2, a 3,a 23, a 4, a 24 will be decoded correctly. Slide ٩ After decoding a 2, a 3,a 23, a 4, a 24, the following vector is subtracted from r The result r () can be expressed as a can be calculated using the following equations Slide ١٠ ٥

6 At the receiver side, a can be decoded using majority decoding from : Similarly, we can decode the following information bits a, a 2, a 3 and a 4. We modify the received vector Slide ١١ In absence of error r (2) is given by a can also be determined from majority logic decoding This is three steps majority logic decoding Slide ١٢ ٦

7 General Muller decdoing To specify the group of check sum, for i < i 2 < i r-l m with l r. the following index set can be formed Let us define E as We form the following set of integers Slide ١٣ General Muller decdoing We have just completed the l th step of decoding the modified received vector is Slide ١٤ ٧

8 Example: Consider RM code of length 6 given in the previous example. If we want to construct the check sum of a 2 (i =, i 2 = 2) The index of the forming sets are Slide ١٥ With l =, the check sum of a 2 is given by Slide ١٦ ٨

9 For a 3, i =. i 2 =3 and The index of the check sum is given by Slide ١٧ We obtain the following check sum for a 3 Follow the same procedure for a 24, a 34 then Now if we want to check a 3 Slide ١٨ ٩

10 The index sets of the check sum are And the check sum are given by Similary we can for the check sums of a, a 2, and a 4. Slide ١٩ performance Slide ٢٠ ١٠

11 Other Construction of Reed Muller code Definition of kroncker product of two matrices A, B a A a 2 a a 2 22 b, B b 2 b b 2 22, For example a A B a 2 B B a a 2 22 B, B Slide ٢١ Krocker product construction The generator matrix for RM(2,4) is knocker product Slide ٢٢ ١١

12 lu lu+v l construction Let u = (u, u,, u n- ) and v = (v, v,, v n- ) be two vectors over GF(2). From u and v we form the following 2n-tuple vector Let C and C 2 are (n, k) code over GF(2) with d 2 > d, we form the following linear code of length 2n Where C is (2n, 2k) linear code, with generator matrix Slide ٢٣ And minimum distance Example: let C be the binary (8,4) linear code of minimum distance 4 generated by And C 2 be (8, ) reptilian code of minimum distance 8 generated by Slide ٢٤ ١٢

13 The resultant code has generator matrix Slide ٢٥ lu lu+v l construction Slide ٢٦ ١٣

14 Squaring construction Consider a binary (n, k) linear code C with generator matrix G. For k k, let C be an (n, k ) linear sub-code of C that is spanned by k rows of G. This portion of C with respect to C is denoted by C/C. Each coset of C is of the following form: l 2 k-k, where for v l, v l is in C but not in C. v l is called leader (representative) of the coset. The vector is representative of C Slide ٢٧ The set of representatives for the cosets in the partition C/C is denoted [C/C] which is called coset representative space for the partition C/C, such that The rows of the generator matrix G is divided k rows that generate C and k k that generate C/C. Let C 2 be an (n, k 2 ) with k 2 k. we can further partition each coset in partition C/C based on C 2 into 2 k -k2 cosets of C 2 ; Each coset consists of the following codeword in C Now we can express C in the following form Slide ٢٨ ١٤

15 let C, C 2,., C m be a sequence of linear subcode of C with dimensions, k, k 2,., k m, respectively, such that: We can form series of partitions Ands C can be expressed in the following form We now can present another method for constructing long codes from a sequence of subcodes. Slide ٢٩ Let C be a binary (n, k ) linear block code with minimum Hamming Distance d, let C, C 2,, C m be a sequence of sub-code of C such that: We form a series of construction like follows Where the generator matrix for each partition has the following property Slide ٣٠ ١٥

16 One level square construction is based on C and the partition C/C. let a = (a, a,, a n- ) and b = (b, b,, b n- ) be two binary n tuples, and let (a, b) denote the 2n tuple (a, a,, a n-,b, b,, b n- ). We form the following set of 2n-tuples Which is a (2n, k + k ) linear block code with minimum Hamming distance. And generator matrix Slide ٣١ Let M and M2 be two matrices with the same number of columns. The matrix is called direct sum and denoted as Then the generator matrix of the squared code can be expressed as Slide ٣٢ ١٦

17 Let M and M2 be two matrices with the same number of columns. The matrix is called direct sum and denoted as Then the generator matrix of the squared code can be expressed as Slide ٣٣ Double square construction Now if we repeat the one-level construction based on V and U/V. This code is a (4n, k + 2k + k2) linear block code with minimum Hamming distance And generator matrix Slide ٣٤ ١٧

18 Which can simplifies as: Or Slide ٣٥ This method can be generalized and can be applied to form longer Reed-Muller Codes if we notice Slide ٣٦ ١٨