Reed Solomon Galois Fields Theory

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The Reed Solomon code is based on the theory of finite fields, named after the French mathematician as Galois fields (GF).

 GALOIS FIELD PROPERTIES A Galois field contains a finite set of elements generated from a primitive element denoted by α where the elements take the values: 0, α0, α1, α2, ..., αN-1where if α is chosen to be 2, N = 2m -1 and m is a predetermined constant. Such a field is described as GF(2m) and contains a set of 2melements. Arithmetic operations (addition, subtraction, multiplication, division) are slightly different in Galois Fields than in the real number system we are used to. This is because any operation (addition, subtraction, multiplication or division) applied in Galois fields must yield results that are elements of the Galois field only. Such a condition cannot be followed through with operations used in the real number system. Galois Fields Addition and Subtraction To describe GF Addition and Subtraction, a useful way to describe the GF elements is in polynomial form. In addition to representing GF elements in index form as above, each of the GF elements may be written as a polynomial of form: am-1xm-1 + ... +a2x2+ a1x1 + a0 where am-1...a0 take values 0 or 1. Thus, a GF element may also be described as a binary number am-1am-2...a1a0. By representing GF elements in binary form, GF addition or subtraction is implemented as the bit-by-bit exclusive-OR function of two binary numbers. Since addition and subtraction are synonymous in the exclusive-OR function of two binary numbers, minus signs in GF arithmetic may always be replaced by plus signs. Before describing GF multiplication and division, let us take a look at where it is applied. Field generator polynomial p(x): Also known as the primitive polynomial, this is an irreducible polynomial (with no factors) of degree m. In this project, the predetermined value for m is 8. The following polynomial is used as the field generator polynomial: x8 + x4 + x3 + x2 + 1. As its name suggests, p(x) is used to construct all the field elements. It does so by using the fact that α is a root of p(x), i.e. p(α) = 0. Thus, α8 + α 4 + α 3 + α 2 + 1 = 0   or      α8= α 4 + α 3 + α 2 + 1* *Minus signs are interchangeable with plus signs in GF arithmetic. Using this equivalent form for α8 , GF elements are generated by multiplying at each stage. The below table displays the first few elements of GF(28) in index form, polynomial form and decimal form. Table 1 Galois field of 256 elements The last GF element is α254 since the maximum element in Galois Fields in αN-1where N = 2m - 1. In our case, m = 8 Galois Fields Multiplication and Division GF Multiplication: Multiplication in Galois field is the product modulo of p(x). GF Division: GF Division is simply long division of GF elements in polynomial form. However subtraction at each stage to find the remainder is GF subtraction (which is the same as GF addition).       Galois Field multiplication and division can also be described conveniently using logarithms. o   Log of GF element is defined as follows: If αx= y then log(y)=x o   Log-inverse of a GF element can be defines as follows: If αx= y then log-1(x)=y o   Then GF multiplication is a●b = log -1( (log (a) + log(b) ) modulo p(x)) o   And GF division is a/b = log -1 ( (log (a) - log(b) ) modulo p(x)) o   a●0 = 0 and a/0 is undefined.﻿﻿

Connecting Galois Fields Arithmetic to Reed Solomon Code

A Reed Solomon code may be described as RS(n,k)  where

This project uses m = 8 bits/symbol and t = 8 symbols as these are the specified DVB-T (Digital Video Broadcasting-Terrestrial) standards. As Galois Field is the basis for Reed Solomon codes, GF(28) is the basis for our Reed Solomon code. A "shortened" RS code is implemented here where n <2m - 1. With n = 32 and t = 8, k = n - 2t or k = 16; so the RS code may be described as RS (32,16). RS codes are fundamentally block codes as opposed to convolutional codes. Block code is when data being transferred is divided into separate blocks. The RS Encoder adds parity symbols to each individual block which results in an output of the form

 data parity data parity data parity ...

Reed Solomon codes are especially helpful for burst errors as all errors in a given symbol are counted only once, i.e. regardless of the number of bits in error in a symbol, RS code treats them as a single error, allowing yet seven more errors to be corrected if t = 8.

REED SOLOMON ENCODING

Our project with m = 8 contains 8-bit symbols and the GF has 28 or a maximum of 256 elements. When the RS Encoder adds 2t parity symbols to the original data of k symbols, 2t + k = n symbols are transmitted. RS Decoder at the receiver is then able to detect up to 2t symbols in error and correct up to t symbols in error. When the locations of symbols in error are not known, RS codes are able to correct up to half the number of parity symbols added. However, when the error locations are previously known (such errors are called erasures), RS codes can correct up to 2t erasures. A combination of errors and erasures may be corrected as long as the sum of these corrections is lesser than the number of parity symbols added.

RS(n,k) can be constructed when both the Encoder and the Decoder agree on a code generator polynomial g(x) which contains 2t factors and is defined as

g(x) =

On the other hand, The Encoder considers the k symbol block as a polynomial M(x) of degree k-1. The Reed Solomon Encoder follows the below steps to construct a T(x) that contains n symbols (k + 2t symbols) for output by the Transmitter.

REED SOLOMON DECODING

Reed-Solomon Decoder considers the incoming message as a polynomial R(x), transmitted message as T(x), and the error introduced as a polynomial E(x). i.e.

Now the Decoder's job is to identify the E(x) so that T(x) can be calculated as follows:

Note that the minus sign is omitted because addition and subtraction are equivalent in GF arithmetic.

Syndromes

From our previous discussion in RS Encoding of how T(x) is formed, we know that T(x) is perfectly divisible by g(x). Thus, in the case that there are no errors for some i in [0, 2t-1], dividing by g(x) must evaluate to zero. However, when there are errors, some or all terms in g(x) will result in a non-zero value. This value is defined as a Syndrome or Si

T() = 0 as x+ is a factor of g(x) which is a factor of T(x)

can be written in the form below where refer to the locations of the errors and the coefficients are the error magnitudes at these locations.

Error locator polynomial

Assuming 'v' is the degree of the error polynomial, the Error Locator Polynomial can be written as:

Roots of the above equation Λ(x) are

Therefore we can write:

Multiplying both sides by

We can create similar error equation for different values of Xj. Terms collected together gives:...

2t-v simultaneous equation can be derived by substituting 'I' with different values in [0, 2t-v-1]. Easiest way to solve the above equation is using Berlekamp's algorithm. Berlekamp's algorithm computes the coefficients of Λ(x) equation using following variables:

•          a correction polynomial C(x) which is initialized to 'x'.
•          syndrome values S0, S1...S2t-1.
•          L, current number of assumed errors; initialized to 0.
•          N, total number of syndromes, main loop of the algorithm will be executed N times. B(x), copy of C(x) when L was last updated; initialized to 1.
•          d, discrepancy in C(x) calculated every iteration.
•          b, copy of d when L was last updated ; initialized to 1. m, number of iteration since L was last updated.
•          m, number of iteration since L was last updated.

Berlekamp's Algorithm

Once the coefficients of Λ(x) are evaluated, finding  is easy because are roots of the equation Λ(x) and we also know that

and j is in [0,n-1], where n is length of code. The Chien search method is applied on Λ(x) to identify Xj eventually for all ej. Chien search method evaluates a given polynomial for all possible roots; in our case the total is n. If equation evaluates to zero, then that value assigned to 'x' is considered as a root.

Prepared by Amulya Kattimani, a Rutgers ECE undergraduate student