Linear Cyclic Codes Problem Set 7 at University of Illinois, Fall 2005 | Assignments Electrical and Electronics Engineering

University of Illinois Fall 2005

ECE 556/CS 577/MATH 579: Problem Set 7

Due: Tuesday October 11, 8:30 a.m.

Reading: Blahut, Algebraic Codes for Data Transmission, Chapters 5 and 6

This Problem Set contains four problems

1. (a) Since every codeword in a cyclic code is a multiple of the generator polynomial g(x), one

form of the syndrome is obtained by dividing the received polynomial R(x) by g(x): the

syndrome is the remainder polynomial S(x), which is, of course, of degree smaller than

deg(g(x)). Prove that for a binary code, the syndrome of the error pattern E(x) = xn−1

is (g(x) + 1)/x.

(b) How many different linear binary cyclic (127,21) codes are there? Why is it not possible

to find a linear binary cyclic (127, 111) code?

2. Let A(z) = 1 + A1z+A2z2+· · · +Anzndenote the weight enumerator of a linear cyclic

binary code Cof length n,nodd, with generator polynomial g(x). Suppose that Ccontains

codewords of odd weight as well as of even weight. Prove that the expurgated code Cexp with

generator polynomial (x+ 1)g(x) contains only the even-weight codewords of C, and that its

weight enumerator is given by [A(z) + A(−z)]/2.

3. Consider a linear binary cyclic t-error-correcting BCH code of length 31.

(a) Find the rates of the codes for the cases 1 ≤t≤8. Are these codes distinct?

(b) What is the BCH bound on the minimum distance of the low rate code whose parity-

check polynomial has αand α3as roots? (This is the dual of a double-error-correcting

BCH code.)

check polynomial has αand α9as roots?

(d) More generally, for odd m, consider the low rate code of length 2m−1 whose parity-check

polynomial has αand α1+2(m+1)/2as roots. Show (as in part (c)) that some conjugate of

α1+2(m+1)/2has a smaller exponent, and find the BCH bound on the minimum distance

of this code.

Codewords from this code are called Gold sequences and are extensively used in commu-

nication systems. Expressed as sequences over the alphabet {+1,−1}instead of {0,1},

their periodic cross-correlations (inner products with cyclic shifts of each other) have

maximum magnitude 1+ 2(m+1)/2which may explain the choice of roots in part (d); you

only need to remember one formula instead of two!

4. Consider a linear binary cyclic t-error-correcting BCH code of length n= 2m−1 whose

generator polynomial has α,α2, . . . , α2tas roots.

(a) Show that the redundancy n−kis at most mt ≈tlog2nand hence the rate is bounded

below by 1 −(log2n)(t/n).

(b) Now consider what happens when nis large. For a channel with fixed error rate p, there

are, on average, np errors, and so we want tto grow proportionally if we increase n.

What happens to the lower bound on the rate?

Note: it can be shown that not just the lower bound, but the rate of a BCH code that

corrects a fixed fraction of errors, goes to 0 as ngrows large.

Long BCH codes are bad . . .

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University of Illinois Fall 2005

ECE 556/CS 577/MATH 579: Problem Set 7

Due: Tuesday October 11, 8:30 a.m. Reading: Blahut, Algebraic Codes for Data Transmission, Chapters 5 and 6

This Problem Set contains four problems

(a) Since every codeword in a cyclic code is a multiple of the generator polynomial g(x), one form of the syndrome is obtained by dividing the received polynomial R(x) by g(x): the syndrome is the remainder polynomial S(x), which is, of course, of degree smaller than deg(g(x)). Prove that for a binary code, the syndrome of the error pattern E(x) = xn−^1 is (g(x) + 1)/x. (b) How many different linear binary cyclic (127, 21) codes are there? Why is it not possible to find a linear binary cyclic (127, 111) code?
Let A(z) = 1 + A 1 z + A 2 z^2 + · · · + Anzn^ denote the weight enumerator of a linear cyclic binary code C of length n, n odd, with generator polynomial g(x). Suppose that C contains codewords of odd weight as well as of even weight. Prove that the expurgated code Cexp with generator polynomial (x + 1)g(x) contains only the even-weight codewords of C, and that its weight enumerator is given by [A(z) + A(−z)]/2.
Consider a linear binary cyclic t-error-correcting BCH code of length 31.

(a) Find the rates of the codes for the cases 1 ≤ t ≤ 8. Are these codes distinct? (b) What is the BCH bound on the minimum distance of the low rate code whose parity- check polynomial has α and α^3 as roots? (This is the dual of a double-error-correcting BCH code.) (c) What is the BCH bound on the minimum distance of the low rate code whose parity- check polynomial has α and α^9 as roots? (d) More generally, for odd m, consider the low rate code of length 2m^ −1 whose parity-check polynomial has α and α1+

(m+1)/ 2 as roots. Show (as in part (c)) that some conjugate of α1+

(m+1)/ 2 has a smaller exponent, and find the BCH bound on the minimum distance of this code. Codewords from this code are called Gold sequences and are extensively used in commu- nication systems. Expressed as sequences over the alphabet {+1, − 1 } instead of { 0 , 1 }, their periodic cross-correlations (inner products with cyclic shifts of each other) have maximum magnitude 1 + 2(m+1)/^2 which may explain the choice of roots in part (d); you only need to remember one formula instead of two!

Consider a linear binary cyclic t-error-correcting BCH code of length n = 2m^ − 1 whose generator polynomial has α, α^2 ,... , α^2 t^ as roots.

(a) Show that the redundancy n − k is at most mt ≈ t log 2 n and hence the rate is bounded below by 1 − (log 2 n)(t/n). (b) Now consider what happens when n is large. For a channel with fixed error rate p, there are, on average, np errors, and so we want t to grow proportionally if we increase n. What happens to the lower bound on the rate? Note: it can be shown that not just the lower bound, but the rate of a BCH code that corrects a fixed fraction of errors, goes to 0 as n grows large. Long BCH codes are bad...

Linear Cyclic Codes Problem Set 7 at University of Illinois, Fall 2005, Assignments of Electrical and Electronics Engineering

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ECE 556/CS 577/MATH 579: Problem Set 7