Codes that attain the Hamming bound are called perfect codes. Examples include codes that have only one codeword, and codes that are the whole of . Another example is given by the repeat codes, where each symbol of the message is repeated an odd fixed number of times to obtain a codeword where q = 2. All of these examples are often called the trivial perfect codes. In 1973, it was proved that any non-trivial perfect code over a prime-power alphabet has the parameters of a Hamming code or a Golay code.
A perfect code may be interpreted as one in which the balls of Hamming radius t centered on codewords exactly fill out the space (t is the covering radius = packing radius). A quasi-perfect code is one in which the balls of Hamming radius t centered on codewords are disjoint and the balls of radius t+1 cover the space, possibly with some overlaps. Another way to say this is that a code is quasi-perfect if its covering radius is one greater than its packing radius.
Read more about this topic: Hamming Bound
Famous quotes containing the words perfect and/or codes:
“If we are on the outside, we assume a conspiracy is the perfect working of a scheme. Silent nameless men with unadorned hearts. A conspiracy is everything that ordinary life is not. It’s the inside game, cold, sure, undistracted, forever closed off to us. We are the flawed ones, the innocents, trying to make some rough sense of the daily jostle. Conspirators have a logic and a daring beyond our reach. All conspiracies are the same taut story of men who find coherence in some criminal act.”
—Don Delillo (b. 1926)
“... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
—Mary Barnett Gilson (1877–?)