Homework 2
ECE 371nhv

Due September 24, 2002 (in class)


(1) Assume that H is the parity check matrix for a linear block code
    C1 with minimum distance d. Assume that d is an odd number. Using
    H obtain a parity check matrix M for a linear block code C2 whose
    minimum distance is d+1. The number of check bits in C2 may be at
    most 2 more than the number of checkbits in C1.

    Explain why you believe your parity check matrix satisfies the
    required property.

(2) Consider the parity check matrix below for a code C3.

        0 1 1 0 1 1 0 1 1 0 1 1
        1 0 1 1 0 1 1 0 1 1 0 1

    A "burst error" of size may introduce errors in any
    e consecutive bits in the transmitted codeword (not all
    e consecutive bits have to be in error, but all errors
    are confined to consecutive e bits).
    
    What is the largest value of e for which code C3 is guaranteed
    to detect e burst errors ?   

(3) Consider the parity check matrix below for a 1-error correcting code.

      0 0 0 1 1 1 1
      0 1 1 0 0 1 1
      1 0 1 0 1 0 1

    If the received word is 1 1 1 0 1 1 1, what will be the decoded
    codeword? Is the decoded codeword identical to the transmitted
    codeword?

(4) Let g(X) be the generator polynomial of a binary cyclic code of
    length n. Show that, if g(X) has X+1 as a factor, the code contains
    no codewordd of odd weight.

    Will the minimum distance of this code be odd or even?

(5) Explain why it is difficult to use CSMA/CD mechanisms in wireless
    networks.

(6) Explain intuitively why slotted ALOHA can yield better performance
    than unslotted ALOHA.