Combinatorial Characterizations of Authentication Codes II |
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Authors: | Rolf S Rees Douglas R Stinson |
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Institution: | (1) Department of Mathematics and Statistics, Memorial University of Newfoundland, A1C 557 St. John's, Newfoundland, Canada;(2) Computer Science and Engineering Department and Center for Communication and Information Science, University of Nebraska-Lincoln, 68588-0115 Lincoln, Nebraska, USA |
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Abstract: | For any authentication code for k source states and v messages having minimum possible deception probabilities (namely, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS% baaSqaaiaadsgaaeqaaOWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0Za% aSGbaeaacaWGRbaabaGaamODaaaaaaa!3F28!\P_d _0 = {k \mathord{\left/ {\vphantom {k v}} \right. \kern-\nulldelimiterspace} v}\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS% baaSqaaiaadsgaaeqaaOWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Ze% xLMBb50ujbqeguuDJXwAKbacfaGae8hkaGIaam4AaiabgkHiTiaaig% dacqWFPaqkcqWFVaWlcqWFOaakcqWF2bGDcqGHsislcaaIXaGae8xk% aKIae8xkaKcaaa!4CD1!\P_d _1 = (k - 1)/(v - 1))\], we show that there must be at least v encoding rules. (This can be thought of as an authentication-code analogue of Fisher's Inequality.) We derive several properties that an extremal code must satisfy, and we characterize the extremal codes for equiprobable source states as arising from symmetric balanced incomplete block designs. We also present an infinite class of extremal codes, in which the source states are not equiprobable, derived from affine planes.Dedicated to Gus Simmons |
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