Page - 444 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 • Throughout theentiresetof28 languagesconsidered in [3], thefirst twoparametersareset to the samevalue1,hence for thepurposeofcomparativeanalysiswithin this family,wecanregarda code like theaboveasa twicespoiledcodeC=C′ 1 f1=(C′′ 2 f2) 1 f1whereboth f1 and f2 areconstantequal to1andC′′ ⊂F42 is thecodeobtainedfromtheabovebycancelingthefirst two letters ineachcodeword. • Conversely, we have C′′ = C′ 2 and C′ = C 1, in terms of the second spoiling operation describedabove. • Toillustrate the thirdspoilingoperation,onecansee, for instance, thatC(0,4)={ 1, 3},while C(1,6)={ 2, 3}. 2.4. TheAsymptoticBound Thespoilingoperationsoncodeswereusedin[13] toprovetheexistenceofanasymptoticbound in thespaceofcodeparameters (δ,R), seealso [16,24,25] formoredetailedpropertiesof theasymptotic bound. LetVq⊂ [0,1]2∩Q2 denote thespaceofcodeparameters (δ,R)ofcodesC⊂Fnq andletUqbethe setofall limitpointsofVq. ThesetUq is characterized in [13]as Uq={(δ,R)∈ [0,1]2 |R≤αq(δ)} foracontinuous,monotonicallydecreasingfunctionαq(δ) (theasymptoticbound).Moreover, code parameters lying inUq arerealizedwith infinitemultiplicity,whilecodepoints inVq\(Vq∩Uq)have finitemultiplicityandcorrespondto the isolatedcodes, see [13,16]. Codes lyingabove theasymptoticboundarecodeswhichhaveextremelygoodtransmissionrate andrelativeminimumdistance,henceverydesirable fromthecodingtheoryperspective. The fact that thecorrespondingcodeparametersarenot limitpointsofothercodeparametersandonlyhavefinite multiplicityreflect the fact thatsuchcodesareverydifficult toreachorapproximate. Isolatedcodes areknowntoarise fromalgebro-geometricconstructions, [14,15]. Relatively little is known about the asymptotic bound: the question of the computability of the function αq(δ)was recently addressed in [25] and the relation toKolmogorov complexitywas investigated in [16]. There are explicit upper and lower bounds for the function αq(δ), see [12], includingthePlotkinbound αq(δ)=0, for δ≥ q−1q ; (5) thesingletonbound,which implies thatR=αq(δ) liesbelowthe lineR+δ=1; theHammingbound αq(δ)≤1−Hq(δ2), (6) whereHq(x) is theq-aryShannonentropy x logq(q−1)−x logq(x)−(1−x) logq(1−x) which is theusualShannonentropyforq=2, H2(x)=−x log2(x)−(1−x) log2(1−x). (7) Onealsohasa lowerboundgivenbytheGilbert–Varshamovbound αq(δ)≥1−Hq(δ) (8) 444