Error Correcting Code

If no two strings in a code differ in fewer than three positions, the we can actually correct a single error, by finding the unique string in the code that differs from the received string in only one position. It turns out that there is a code of 7-bit strings that corrects single errors and contains 16 strings. Find such a code. Hint: Reasoning it out is probably best, but if you get stuck, write a program that searches for the string. From the example in the book - the strings would be like ASCII sequence of bits - whereby ASCII has a seven bit code and then a parity bit placed on the front for error detecting.

Recursive definition

We can define sorted lists of integers as follows: BASIS - A list consisting of a single integer is sorted. INDUCTION - If L is a sorted list in which the last element is a and if b >= a, then L followed by b is a sorted list. Prove that this recursive definition of "sorted list" is equivalent to our original, nonrecursive definition, which is that the list consist of integers a1 <= a2 <= .... <= an Remember, you need to prove to parts: (a) If a list is sorted by the recursive definition, then it is sorted by the nonrecursive definition (b) if a list is sorted by the nonrecursive definition, then it is sorted by the recursive definition. Part(a) can use induction on... click for more

Induction by Recursion : Even-Parity Strings

Define recursively the set of even-parity strings, by induction on the length of the string. Hint: It helps to define two concepts simultaneously, both the even-parity and odd-parity strings.

Discrete Math : Subsets and Elements

Let S = {1,2,3,4,5,6,7,8}. Determine: (a) The number of subsets of S (b) The number of subsets of S with at most four elements (c) The number of ordered lists with elements chosen form S (with possible repititions) (d) The number of ordered lists with nine elements chosen form S with no repititions (e) The number of element multisets chosen from S.

Discrete

Let Pn be the product of the first n odd numbers... (see attachment)