On McMillans theorem about uniquely decipherable codes

On McMillan’s theorem ab out uniquely decipherable co des Stephan F oldes T amp e re Univ ersit y of T ec hnology PL 553, 33101 T amp e re, Finland sf@tut.fi 19 June 2008 Abstract Karush’s pr o of of McMil lan ’s the or em is r e c ast as an ar gument involving p olynomials with non-c om mut ing indeterminates c ertain evaluations of w hich yield the Kr aft sums o f c o des, pr oving a str engthene d version of McMil lan ’s the or em. Let len : A ∗ − → N be the length function on the free monoid of all strings o ver a giv en no n- em pt y finite set A. Let con : A ∗∗ − → A ∗ b e the concatena- tion map whic h to ev ery string of strings asso ciates their concatenation. A uniquely de cipher able c o de is a finite set C ⊆ A ∗ suc h that con is injectiv e on t he submonoid C ∗ of A ∗∗ . T his submonoid is then isomorphic t o the sub- monoid C = con [ C ∗ ] of A ∗ freely generated b y C. Denoting by r the num b er of elemen ts of the alp hab et A , the Kr aft sum K ( C ) of an y finite C ⊆ A ∗ is defined as P x ∈ C r − len ( x ) . In [M] McMillan sho w ed that if C is a uniquely decipherable co de, then its K raft sum is a t most 1 . The pro of usually giv en is that of Karush [K ]. This pro o f can b e recast as an argumen t in v olving ev alu- ations of p o ly nomials with non-comm uting indeterminates corresp onding to the v ario us (infinitely man y) strings in A ∗ , as follow s. Let R h A ∗ i b e the free asso ciativ e R -algebra g e nerated b y the elemen ts of A ∗ considered a s indeterminates, i.e. R h A ∗ i is the non-comm ut a tiv e ring of formal p olynomials with real co efficien ts in the non-commuting indeter- minates x ∈ A ∗ . F o r w = ( x 1 , ..., x n ) in A ∗∗ let P ( w ) denote the mono mia l x 1 · ... · x n in R h A ∗ i . Let C and D b e finite uniquely decipherable co des o v er a non-empt y finite alphab et A with r elemen ts, and supp ose tha t C ⊆ D. The Kraft sum K ( C ) of C is then t he ev aluation of the p olynomial P x ∈ C x at x := r − len ( x ) for x ∈ A ∗ . 1 Fix a p ositiv e integer k . F or an y p ositiv e integer l , part it ion the set D l in to tw o disjoin t sets: W l 1 =  w ∈ D l : con ( w ) ∈ con [ C k ]  W l 2 = D l \ W l 1 F or eve ry l the p olynomial  P x ∈ D x  l = P w ∈ D l P ( w ) (1) is equal to the sum P w ∈ W l 1 P ( w ) + P w ∈ W l 2 P ( w ) (2) Let m b e the largest in teger n with C ∩ con [ D n ] 6 = ∅ . Then the p olynomial mk P l = k  P x ∈ D x  l = mk P l = k P w ∈ D l P ( w ) (3) is the sum of mk P l = k P w ∈ W l 1 P ( w ) (4) and mk P l = k P w ∈ W l 2 P ( w ) (5) Let I ( C , D ) b e the ideal of R h A ∗ i generated b y the p olynomials x − P ( w ) for x ∈ C , w ∈ D ∗ , x = co n ( w ) . Mo dulo this ideal, (4) is congruen t to  P x ∈ C x  k (6) The ho m omorphism R h A ∗ i − → R ev aluating each p olynomial at x := r − len ( x ) is null on the ideal I ( C , D ) and therefore the ev aluation of (3) equals 2 the sum of the ev aluations of (5) and (6 ). The ev aluation of (5) b eing non- negativ e, the ev alua t io n of (6) is at most the ev aluation of (3). F or the Kraft sums K ( C ) and K ( D ) this means that K ( C ) k ≤ mk P l = k K ( D ) l (7) Applying this to D = A 1 , as C ⊆ A 1 and obviously K ( A 1 ) = 1 , w e obtain K ( C ) k ≤ mk − k + 1 ≤ mk and hence K ( C ) k ≤ 1 and K ( C ) ≤ 1 for all uniquely decipherable co des C . This holds for all k ≥ 1 . Recom bining this with (7), letting C and D b e arbitrary finite uniquely decipherable co des with C ⊆ D , and using no w the kno wledge that K ( D ) ≤ 1, w e o bta in K ( C ) k ≤ mk P l = k K ( D ) l ≤ mk · K ( D ) k (8) Recall that the definition of m is indep endent of the c hoice o f k . Thus (8 ), b eing true for all k ≥ 1, yields the inequalit y K ( C ) ≤ K ( D ) : Extended McMillan Theorem If C an d D ar e uniquely de cipher able c o des over the sa me alphab et, such that every string in C is a c onc atenation of strings in D , then the Kr aft sum of C is les s then or e qual to the Kr aft sum of D . This statement clearly includes the classical McMillan Theorem, corre- sp onding to the case where D consists of all strings of length 1 . References [K] J. Karush, A simple pro of of a n inequalit y of McMillan, IRE T rans. Information Theory IT-7 (1961) 118-118 [M] B. McMillan, Tw o inequalities implied b y unique decipherabilit y , IRE T rans. Informat ion Theory IT-2 (19 56) 115-116 3