Minimum distance bounds for error-correcting codes

Communication is the transmission of information from a source to a receiver (destination). Human speech, telephone conversations, high frequency radios, and space communication links each involve communication. In each case information is passed from a source through a channel (telephone line, space, etc.) to a receiver. If the channel is "noiseless" the information being transmitted is not altered and is received correctly. However, most channels are noisy, where noise is defined as any alteration of the message in a non-deterministic or probabilistic way. Messages transmitted through noisy channels may be affected by the noise, resulting in a different message being received. For example, cross-talk in telephone conversations, lightning, or static may cause errors to be introduced in the transmitted message. The communication problem is to determine a way to decrease the effects of the noise in the channel on the message so that the message may be transmitted as reliably as possible. In this thesis it will be assumed that information is represented as binary numbers. For example, a letter of the alphabet may be coded as the binary number which denotes the letter's position in the alphabet. The letter m is the thirteenth letter of the alphabet and is coded as 01101. Any English message is converted into this code before being transmitted and the coded message is converted back to English when received, as in the following model