AN2407 查看數據表（PDF） - Freescale Semiconductor

零件编号

产品描述 (功能)

生产厂家

AN2407

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples

Freescale Semiconductor 

Theory

where i ∈ N , can produce 2m–1 field elements (excluding the zero element). In general, extended Galois fields of

class GF(2m) possess 2m elements, where m is the symbol size, that is, the size of an element, in bits. For example,

in ADSL systems, the Galois field is GF(256). It is generated by the following primitive polynomial:

1+x2+x3+x4+x8

This is a degree-eight irreducible polynomial. The field elements are degree-seven polynomials. Due to the one-to-

one mapping that exists between polynomials over GF(2) and binary numbers, the field elements are representable

as binary numbers of eight bits each, that is, as bytes. In GF(2m) fields, all elements besides the zero element can be

represented in two alternative ways:

1. In binary form, as an ordinary binary number.

2. In exponential form, as αp. It follows from these definitions that the exponent p is an integer ranging

from 0 to (2m–2). Conventionally, the primitive element is chosen as 0x02, in binary representation.

As for GF(2), addition over GF(2m) is the bitwise XOR of two elements. Galois multiplication is performed in two

steps: multiplying the two operands represented as polynomials and taking the remainder of the division by the

primitive polynomial, all over GF(2). Alternatively, multiplication can be performed by adding the exponents of

the two operands. The exponent of the product is the sum of exponents, modulo 2m –1.

Polynomials over the Galois field are of cardinal importance in the Reed-Solomon algorithm. The mapping

between bitstreams and polynomials for GF(2m) is analogous to that of GF(2). A polynomial of degree D over

GF(2m) has the most general form:

f(x) = f0 + f1x + f2x2 + f3x3… + fDxD

where the coefficients f0 – fD are elements of GF(2m). A bitstream of (N+1)m bits is mapped into an abstract

polynomial of degree N by setting the coefficients equal to the symbol values and the exponents of x equal to the bit

locations. The Galois field is GF(256), so the bitstream is divided into symbols of eight consecutive bits each. The

first symbol in the bitstream is 00000001. In exponential representation, 00000001 becomes α0. Thus, α0 becomes

the coefficient of x0. The second symbol is 11001100, so the coefficient of x is α127 and so on.

... 11110011

11111101

10110111

01110101

11001100

00000001

α233

α80

α158

α21

α127

α0

↔ f(x) = α0 + α127 x + α21 x2 + α158 x3 + α80 x4 + α233 x5

The elements are conventionally arranged in a log table so that the index equals the exponent, and the entry equals

the element in its binary form. Table 1 displays the log table for ADSL systems.

Table 1. Exponential-to-Binary Table for ADSL Systems

p

αp

0

0x01

1

0x02

2

0x04

3

0x08

4

0x10

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples, Rev. 1

Freescale Semiconductor

5

Share Link: