In this paper we study authentication codes (A-codes) that are r-fold secure against spoofing. We obtain necessary and sufficient conditions for such codes and give a characterization of codes with a minimum number of encoding rules. We will show that if the source is uniform, A-codes that provide perfect protection against spoofing attack of order r are r-fold secure.

The Pi code constructed by V.I. Panchenko (1987) is studied. The Pi code as an alternative to the Hamming code in the class of single-error-correcting and double-error-detecting codes (SEC-DED codes) is also considered. The Pi code has a smaller number of words of weight 4 and provides a larger probability of triple-independent-error detection than the shortened Hamming code with the same parameters. Shortening algorithms for the Pi code are proposed, and parity check matrices of the (39,32), (72,64), (137,128) shortened Pi codes are constructed. The obtained codes can detect byte errors of length 4. The parity check matrices of the Pi code have more 1's in rows than corresponding matrices of the Hamming code. The Pi code is a reasonable alternative to the Hamming code in the class of SEC-DED codes.>

Infinite families of linear binary codes with covering radius R=2 and minimum distance d=3 and d=4 are given. Using the constructed codes with d=3, R=2, families of covering codes with R>2 are obtained. The parameters of many constructed codes with R>