Sharper upper bounds for $q$-ary $B_2$ codes from Toeplitz SDPs

Sharper upper bounds for q -ary B 2 codes from T oeplitz SDPs Stefano Della Fiore Department of Information Engineering University of Br escia Brescia, Italy stefano.dellafiore@unibs.it Abstract —In a recent note we derived information theoretic upper bounds on the rate of q -ary 2 -separable codes and on the rate of q -ary B 2 codes based on an entropy argument applied coordinate-wise to a suitable pair of random suffixes of codewords. In the B 2 case, the bound was obtained by maximizing the entropy of the differ ence X − Y of two independent q -ary random v ariables under the sole constraint P ( X = Y ) ≥ 1 /q . In this paper we refine this step by exploiting the full Fourier -analytic structure of the difference distrib ution X − Y . Mor e precisely , we use that the pmf of X − Y is an autocorrelation of a probability mass function on { 0 , . . . , q − 1 } and theref ore its Fourier transform is a nonnegative trigonometric polynomial of prescribed degree. This leads to a natural con vex optimization pr oblem over the coefficients of such polynomials whose optimal value yields a strictly smaller upper bound on the entropy of X − Y and, in turn, to improved bounds on the rate of q -ary B 2 codes. W e ev aluate the resulting bound numerically via truncated T oeplitz SDPs and show that for q ∈ { 9 , 10 , 11 , 12 , 13 } the new rate upper bounds improve upon the best available bounds in the literatur e. Index T erms — B 2 codes, Fourier methods, semidefinite pro- gramming I . I N T RO D U C T I O N Let [0 , q − 1] = { 0 , 1 , . . . , q − 1 } and let C n ⊂ [0 , q − 1] n be a q -ary code of length n with | C n | = M codewords. W e define the asymptotic base- q rate of a f amily { C n } as R = lim sup n →∞ 1 n log q | C n | , where logarithms without subscript are to base 2 . Definition 1.1: A q -ary code C n = { c 1 , . . . , c M } ⊂ [0 , q − 1] n is a B 2 code if all sums (ov er the reals) c i + c j , 1 ≤ i ≤ j ≤ M , are distinct. For q = 2 this definition is equiv alent to the usual definition of a ¯ 2 -separable code. Bounds on the rate of q -ary B 2 codes hav e been given in particular in [2]–[4], [6], [7], see also the references therein. Related entropy/prefix–suffix methods for (binary) constant-weight B 2 sequences were studied recently in [8]. A con venient information-theoretic route to rate upper bounds is to associate to a code a random pair of code words (or suffix es of codew ords), and relate the code size to entropies of suitably chosen random variables. For B 2 codes, a k ey step is to control the entropy of the dif ference X − Y of two i.i.d. q -ary random variables supported on [0 , q − 1] (the difference being ov er the integers). In our earlier w ork [5], a coarse relaxation only used the following constraint P ( X = Y ) = q − 1 X a =0 P ( X = a ) 2 ≥ 1 q , leading to a closed-form upper bound on H( X − Y ) and thus to an explicit rate upper bound. In this paper we strengthen that entropy step by exploiting the full Fourier -analytic structure of X − Y . Indeed, the characteristic function of X − Y satisfies ψ X − Y ( θ ) = | ϕ ( θ ) | 2 , where ϕ ( θ ) = E [ e iθX ] . Equiv alently , the pmf of X − Y is the autocorrelation of a pmf on [0 , q − 1] , and its Fourier series is a nonnegati ve trigonometric polynomial of de gree at most q − 1 . This yields a conv ex optimization problem o ver dif ference pmfs (still a relaxation, since we do not enforce the existence of a spectral factor with nonnegativ e coefficients summing to one), and the same feasible set can be described by an infinite family of T oeplitz positi ve semidefinite (PSD) constraints. W e make this equiv alence explicit and pro vide standard references. W e then provide a self-contained deriv ation showing ho w the refined bound on H( X − Y ) improves the rate upper bound for q -ary B 2 codes. Finally , we ev aluate the new bound numeri- cally via truncated T oeplitz semidefinite programs (SDPs): for q ∈ { 9 , 10 , 11 , 12 , 13 } , the resulting rate upper bounds improv e upon the strongest bounds we are aware of in the literature, including [3], [5]–[7] (see Section V). The note is organized as follows. In Section II we recall the Fourier representation of the difference distribution X − Y and deriv e the corresponding nonnegati vity constraints. In Section III we formulate an associated entropy maximization problem and discuss its equiv alent T oeplitz SDP formulation. In Section IV we show how this improv ed entropy bound enters the proof of the rate bound for B 2 codes, and we state our main result. Section V contains numerical e valuations and comparisons with existing literature bounds. Contributions. (i) W e introduce a Fourier -analytic relaxation for the i.i.d. dif ference distribution D = X − Y that ex- ploits the fact that its characteristic function is a nonnegati ve trigonometric polynomial of degree at most q − 1 . (ii) W e show that this relaxation is equiv alently characterized by an infinite family of T oeplitz PSD constraints, making explicit the link with classical positi ve-definiteness results for Fourier coefficients. (iii) Plugging the refined entropy bound into the prefix–suffix method yields a strictly impro ved asymptotic upper bound on the rate of q -ary B 2 codes. (iv) W e ev aluate the bound numerically via truncated T oeplitz SDPs and obtain new best-known (to our knowledge) rate upper bounds for q ∈ { 9 , 10 , 11 , 12 , 13 } . I I . F O U R I E R R E P R E S E N TA T I O N O F T H E D I FF E R E N C E D I S T R I B U T I O N Let X be a random variable with values in [0 , q − 1] and pmf P = ( p 0 , . . . , p q − 1 ) , and let Y be an independent copy of X . W e are interested in the pmf of the dif ference D = X − Y , where the subtraction is performed ov er the integers. Then D takes values in {− q + 1 , . . . , q − 1 } and its probability mass function r = ( r − q +1 , . . . , r q − 1 ) is giv en by r k = P ( D = k ) = q − 1 X i =0 p i p i + k , k = − ( q − 1) , . . . , q − 1 , (1) with the con vention p j = 0 for j / ∈ { 0 , . . . , q − 1 } . W e now introduce the characteristic function of X : ϕ ( θ ) = E [ e iθX ] = q − 1 X j =0 p j e iθj , θ ∈ [0 , 2 π ] . The characteristic function of D = X − Y is then ψ D ( θ ) = E [ e iθ ( X − Y ) ] = ϕ ( θ ) ϕ ( θ ) = | ϕ ( θ ) | 2 . (2) On the other hand, by definition of Fourier series we ha ve ψ D ( θ ) = q − 1 X k = − ( q − 1) r k e ikθ =: R ( θ ) , (3) where R ( θ ) is a trigonometric polynomial of degree at most q − 1 with real, symmetric coefficients r k = r − k . Combining (2) and (3) we obtain the representation R ( θ ) = | ϕ ( θ ) | 2 ≥ 0 , θ ∈ [0 , 2 π ] . (4) Moreov er, the coefficients of R ( θ ) can be recovered from ϕ ( θ ) by the standard Fourier in version formula r k = 1 2 π Z 2 π 0 R ( θ ) e − ikθ dθ = 1 2 π Z 2 π 0 | ϕ ( θ ) | 2 e − ikθ dθ , | k | ≤ q − 1 . (5) It is con venient to single out the coefficient r 0 . From either (1) or (5) we hav e r 0 = P ( X = Y ) = q − 1 X i =0 p 2 i = 1 2 π Z 2 π 0 | ϕ ( θ ) | 2 dθ . By Cauchy–Schwarz, q − 1 X i =0 p 2 i ≥ 1 q  q − 1 X i =0 p i  2 = 1 q , so that r 0 = P ( X = Y ) ≥ 1 q . (6) Remark 2.1: The existence of ϕ and of a nonnegati ve trigonometric polynomial R with coefficients r k is closely related to classical results such as Herglotz’ s theorem and Fej ´ er–Riesz factorization; see e.g. [1], [9], [10]. I I I . E N T RO P Y B O U N D S V I A N O N N E G A T I V E T R I G O N O M E T R I C P O LY N O M I A L S Let X , Y be i.i.d. random variables with pmf P on [0 , q − 1] and let D = X − Y as abov e. W e denote by h ⋆ q := sup H( D ) , where the supremum is tak en ov er all such P , and H denotes Shannon entropy . In [5] we upper bounded h ⋆ q by H( D ) ≤ ¯ h q := H  1 q , 1 2 q , . . . , 1 2 q | {z } 2 q − 2 times  = 1 q log q + q − 1 q log(2 q ) , (7) where the inequality is obtained by maximizing the entropy under the sole constraint P ( D = 0) ≥ 1 /q . W e adopt the standard conv ention 0 log 0 := 0 . The Fourier-analytic description suggests the following con- ve x optimization problem ov er the difference distributions. Definition 3.1: For fixed q ≥ 2 define h F q as the optimal value of maximize − q − 1 X k = − ( q − 1) r k log r k subject to r k ≥ 0 , q − 1 X k = − ( q − 1) r k = 1 , r − k = r k , r 0 ≥ 1 q , R ( θ ) := q − 1 X k = − ( q − 1) r k e ikθ ≥ 0 for all θ ∈ [0 , 2 π ] . (8) Lemma 3.2: For ev ery q ≥ 2 we ha ve h ⋆ q ≤ h F q ≤ ¯ h q . Pr oof: Any difference distribution r arising from D = X − Y with X , Y i.i.d. and X ∈ [0 , q − 1] satisfies r k ≥ 0 , P k r k = 1 , symmetry r − k = r k , and r 0 ≥ 1 /q by (6). Moreov er, its trigonometric polynomial R satisfies R ( θ ) = | ϕ ( θ ) | 2 ≥ 0 by (4). Hence such an r is feasible for (8), gi ving h ⋆ q ≤ h F q . Dropping the constraint R ( θ ) ≥ 0 enlarges the feasible set and yields exactly (7), hence h F q ≤ ¯ h q . W e now make explicit the connection between the Fourier formulation (8) and a semidefinite-programming formulation based on T oeplitz matrices. Definition 3.3 (T oeplitz SDP formulation): Extend the se- quence ( r k ) by setting r k = 0 for | k | ≥ q . For N ≥ 1 define the T oeplitz matrix T ( N ) ( r ) by T ( N ) ( r ) ij = r i − j , i, j ∈ { 0 , . . . , N − 1 } . For fixed q ≥ 2 define h SDP q as the optimal value of maximize − q − 1 X k = − ( q − 1) r k log r k subject to r k ≥ 0 , q − 1 X k = − ( q − 1) r k = 1 , r − k = r k , r 0 ≥ 1 q , T ( N ) ( r ) ⪰ 0 for all N ≥ 1 . (9) Pr oposition 3.4: For ev ery q ≥ 2 we ha ve h F q = h SDP q . Pr oof: W e sho w that the feasibility conditions in (8) and (9) are equiv alent. ( R ( θ ) ≥ 0 ⇒ T ( N ) ( r ) ⪰ 0 ). Fix N and c = ( c 0 , . . . , c N − 1 ) ∈ C N , and define the trigonometric polynomial p c ( θ ) = N − 1 X j =0 c j e ij θ . Using that r k are the Fourier coef ficients of R ( θ ) , a standard identity (see e.g. [9, Ch. 1]) giv es c ∗ T ( N ) ( r ) c = N − 1 X i,j =0 c i c j r i − j = 1 2 π Z 2 π 0 R ( θ ) | p c ( θ ) | 2 dθ . If R ( θ ) ≥ 0 for all θ , the integral is nonne gative for every c , hence T ( N ) ( r ) ⪰ 0 . ( T ( N ) ( r ) ⪰ 0 ⇒ R ( θ ) ≥ 0 ). Assume T ( N ) ( r ) ⪰ 0 for all N . Fix θ 0 ∈ [0 , 2 π ] and for m ≥ 0 define p m ( θ ) = m X j =0 e ij ( θ − θ 0 ) . Let c ( m ) be its coefficient vector (so N = m + 1 ). By the identity above and T ( m +1) ( r ) ⪰ 0 we have 0 ≤ 1 2 π Z 2 π 0 R ( θ ) | p m ( θ ) | 2 dθ . Define the Fej ´ er kernel by F m +1 ( θ ) := 1 m + 1    m X j =0 e ij θ    2 , so that F m +1 ≥ 0 and 1 2 π R 2 π 0 F m +1 ( θ ) dθ = 1 . Then | p m ( θ ) | 2 = ( m + 1) F m +1 ( θ − θ 0 ) , and therefore 0 ≤ ( R ∗ F m +1 )( θ 0 ) . Since R is continuous and ( F m ) is an approximate identity , we hav e ( R ∗ F m +1 )( θ 0 ) → R ( θ 0 ) as m → ∞ (see e.g. [10, Ch. 2]). Therefore R ( θ 0 ) ≥ 0 . Since θ 0 is arbitrary , R ( θ ) ≥ 0 for all θ . Thus the feasible sets coincide, and the objecti ve functions are identical, so the optimal values agree: h F q = h SDP q . I V . I M P R OV E D B O U N D S F O R B 2 C O D E S W e no w show ho w the improv ed entropy bound h F q for D = X − Y enters the deriv ation of an upper bound on the rate of q -ary B 2 codes. The proof belo w follows the same structure as in [5] and is self-contained. Let C n be a q -ary B 2 code of length n with M = | C n | codew ords. Split each codeword into a prefix of length e and a suffix of length f = n − e , and partition C n into classes P 1 , . . . , P r according to the prefix. Let M i = | P i | and denote by ℓ i ∈ [0 , q − 1] e the common prefix of P i . Thus each class can be written as P i = { ( ℓ i , w ) : w ∈ W i } , | W i | = M i . Let ( X , Y ) be a random ordered pair of suf fixes obtained by choosing uniformly an ordered pair of code words inside the same prefix class, and then taking their suf fixes. Then ( X , Y ) is uniform ov er P r i =1 M 2 i possibilities. Hence H( X, Y ) = log r X i =1 M 2 i . Lemma 4.1: If C n is a B 2 code, then the map Φ( w 1 , w 2 ) = w 1 − w 2 is injectiv e on ordered pairs of distinct suffixes (tak en within a prefix class). In other w ords, the multiset of differences { w 1 − w 2 | ( w 1 , w 2 ) ∈ W i × W i , w 1  = w 2 , i = 1 , . . . , r } contains no repetitions. Pr oof: W e provide a short self-contained proof in the same spirit as in [5]. Assume, for the sake of contradiction, that there exist four distinct code words ( ℓ h , w 1 ) , ( ℓ h , w 2 ) ∈ P h , ( ℓ m , w 3 ) , ( ℓ m , w 4 ) ∈ P m , such that Φ( w 1 , w 2 ) = Φ( w 3 , w 4 ) , (10) either with h = m and ( w 1 , w 2 )  = ( w 3 , w 4 ) , or with h  = m . Case 1: h = m . If { w 1 , w 2 } ∩ { w 3 , w 4 }  = ∅ , then (10) either forces ( w 1 , w 2 ) = ( w 3 , w 4 ) or contradicts the B 2 property of the code C n . Otherwise { w 1 , w 2 } ∩ { w 3 , w 4 } = ∅ . From (10) we hav e w 1 − w 2 = w 3 − w 4 , hence w 1 + w 4 = w 2 + w 3 . Therefore ( ℓ h , w 1 ) + ( ℓ h , w 4 ) = (2 ℓ h , w 1 + w 4 ) = (2 ℓ h , w 2 + w 3 ) = ( ℓ h , w 2 ) + ( ℓ h , w 3 ) . (11) The two unordered pairs of code words { ( ℓ h , w 1 ) , ( ℓ h , w 4 ) } and { ( ℓ h , w 2 ) , ( ℓ h , w 3 ) } are distinct (the four codew ords are distinct), yet (11) shows they hav e the same sum, contradicting the B 2 property . Case 2: h  = m . From (10) we again obtain w 1 + w 4 = w 2 + w 3 , hence ( ℓ h , w 1 ) + ( ℓ m , w 4 ) = ( ℓ h + ℓ m , w 1 + w 4 ) = ( ℓ h + ℓ m , w 2 + w 3 ) = ( ℓ h , w 2 ) + ( ℓ m , w 3 ) . Thus two distinct unordered pairs of code words have the same sum, again contradicting the B 2 property . In both cases we reach a contradiction. Let Z := X − Y ∈ Z f denote the componentwise dif ference of the tw o suffix es (ov er the integers), i.e., Z = ( Z 1 , . . . , Z f ) with Z j = X j − Y j . From Lemma 4.1 we obtain the entropy decomposition H( X, Y ) = H( Z ) + P ( Z = 0) H( X , Y | Z = 0) . Lemma 4.2: Let Z = X − Y be the suffix dif ference defined abov e. Then P ( Z = 0) H( X, Y | Z = 0) ≤ r M log M . In particular, with the choice e = j log q (2 M ) − log q log M k , r = q e , (12) we have that P ( Z = 0) H( X , Y | Z = 0) ≤ 2 . Pr oof: Let S 2 := P r i =1 | P i | 2 . By construction, ( X , Y ) is uniform over the S 2 ordered pairs of suffix es drawn within the same prefix class, hence each such ordered pair has probability 1 /S 2 . The ev ent Z = 0 is equiv alent to X = Y , which corresponds exactly to the M = P r i =1 | P i | diagonal ordered pairs (one per codew ord). Therefore P ( Z = 0) = P ( X = Y ) = M S 2 . (13) By Cauchy–Schwarz, S 2 = r X i =1 | P i | 2 ≥  P r i =1 | P i |  2 r = M 2 r , hence from (13) we obtain P ( Z = 0) ≤ r / M . Moreov er, since all diagonal pairs hav e the same probability 1 /S 2 , the conditional distrib ution of ( X , Y ) giv en Z = 0 is uniform over its support, which has size M . Thus H( X, Y | Z = 0) = log M . Finally , with (12) we have r = q e ≤ 2 M / log M , and therefore P ( Z = 0) H( X, Y | Z = 0) ≤ r M log M ≤ 2 . Write Z = ( Z 1 , . . . , Z f ) . By subadditivity , H( Z ) ≤ f X j =1 H( Z j ) . Fix j ∈ { 1 , . . . , f } . Let r ( j ) be the pmf of Z j = X j − Y j . Let I ∈ { 1 , . . . , r } denote the random inde x of the prefix class from which the ordered pair of codewords is drawn. Conditioned on the prefix class index I = i , the pair ( X j , Y j ) is i.i.d. on [0 , q − 1] , hence the conditional difference pmf r ( j ) | I = i is feasible for (8). Since the feasible set of (8) is con ve x and r ( j ) = P i P ( I = i ) r ( j ) | I = i is a conv ex combination of these conditional pmfs, it follo ws that r ( j ) is feasible for (8) as well. Therefore H( Z j ) ≤ h F q , and consequently H( Z ) ≤ f X j =1 H( Z j ) ≤ f h F q . W e now fix e as in (12). Then Lemma 4.2 giv es P ( Z = 0) H( X , Y | Z = 0) ≤ 2 , and hence log r X i =1 | P i | 2 = H( X, Y ) ≤ f h F q + 2 . (14) On the other hand, the Cauchy–Schwarz inequality yields M 2 r ≤ r X i =1 | P i | 2 . (15) Using (15) and (14) we obtain 2 log M − log r ≤ f h F q + 2 . W ith the choice (12) we ha ve log r = e log q = log(2 M ) − log log M + O (1) , hence 2 log M − log r = log M + log log M + O (1) . (16) Moreov er, (12) also giv es e = log M log q + O (log log M ) , f = n − e = n − log M log q + O (log log M ) . (17) Substituting (16)–(17) into 2 log M − log r ≤ f h F q + 2 and absorbing constants yields  1 + h F q log q  log M ≤ n h F q + O (log log M ) . T ABLE I N U M E R I C A L C O M PA R I S O N O F R ATE U P P E R B O U N D S F O R q - A RY B 2 C O D E S . A L L N U M B E R S A R E RO U N D E D U P W A R D S . q = 9 10 11 12 13 R new q 0.55792 0.55611 0.55457 0.55323 0.55206 Lindstr ¨ om [3] 0.57551 0.56839 0.56264 0.55789 0.55390 Ours [5] 0.56149 0.55966 0.55807 0.55668 0.55545 W ang [7] 0.55841 0.55727 0.55626 0.55536 0.55454 Dividing by n and using log log M = o ( n ) (since M ≤ q n ) we obtain 1 n log M ≤ h F q log q log q + h F q + o (1) . Theor em 4.3: For integer q ≥ 2 , let R b be the asymptotic rate of q -ary B 2 codes. Then R b ≤ h F q log q + h F q , where h F q is the optimal v alue of the Fourier -analytic con vex program (8), equi valently of the T oeplitz semidefinite pro- gram (9). V . N U M E R I C A L E V A L UAT I O N A N D C O M PA R I S O N W I T H L I T E R A T U R E B O U N D S This section reports numerical upper bounds obtained from the truncated T oeplitz SDP formulation of Section III, and compares the resulting rate bounds with previously kno wn upper bounds in the literature. A. Computation of h F q via truncated T oeplitz SDPs For each q we solv e a truncated v ersion of (9) by imposing the semidefinite constraints T ( N ) ( r ) ⪰ 0 only for N ≤ N max . Let h q ,N max denote the resulting optimal value (in bits). Then h ⋆ q ≤ h F q = h SDP q ≤ h q ,N max , so that h q ,N max provides a certified upper bound on h ⋆ q . W e report the values obtained for N max = 100 . Giv en h q ,N max , Theorem 4.3 yields the rate upper bound R b ≤ R new q := h q ,N max log q + h q ,N max . B. Comparison with known bounds W e compare R new q against the best a vailable upper bounds on the rate of q -ary B 2 codes from: Lindstr ¨ om [3], Della Fiore– Dalai [5], and W ang [7]. For q ∈ { 9 , 10 , 11 , 12 , 13 } , the ne w bounds improve upon all of the abov e. T able I summarizes the values. C. Implementation details and numerical stability The truncated program is a concave maximization over a spectrahedral set. In practice, we solve it using a standard conic formulation of the entropy term (e.g., via the relativ e- entropy/exponential cone), together with Hermitian PSD con- straints on the T oeplitz matrices T ( N ) ( r ) for N ≤ N max . W e enforce the symmetry r − k = r k explicitly and optimize ov er the 2 q − 1 v ariables ( r − ( q − 1) , . . . , r q − 1 ) . T o assess numerical stability , one can v erify that the resulting upper bounds h q ,N max (and hence R new q ) stabilize as N max increases, e.g., by comparing N max ∈ { 50 , 100 , 150 } . As expected from the monotonicity of the relaxation, the sequence h q ,N max is nonincreasing in N max and provides certified upper bounds for ev ery finite truncation le vel. V I . D I S C U S S I O N A N D O U T L O O K A. On the r elaxation gap: positivity vs. spectral factorization The feasible set of (8) consists of symmetric pmfs r on {− ( q − 1) , . . . , q − 1 } whose trigonometric polynomial R ( θ ) is nonnegati ve on the unit circle and satisfies r 0 ≥ 1 /q . Ev ery i.i.d. difference distribution D = X − Y generates such an r with R ( θ ) = | ϕ ( θ ) | 2 , where ϕ is a degree- ( q − 1) polynomial with nonnegativ e coef ficients summing to one. In (8) we only enforce the nonnegati vity of R ( θ ) , i.e., the existence of some spectral factor in L 2 , but we do not impose that R ( θ ) admits a factorization with nonnegative time-domain coefficients summing to one. Thus, the program provides a principled relaxation that captures all Fourier posi- tivity constraints of i.i.d. differences while remaining conv ex. Understanding when the relaxation is tight (or how large the gap can be) is an interesting direction for future work. B. Why F ourier structur e impr oves o ver the collision constraint The bound of [5] only uses the collision probability con- straint r 0 = P ( X = Y ) ≥ 1 /q . In contrast, the Fourier constraint R ( θ ) ≥ 0 enforces that the whole sequence ( r k ) is positiv e definite in the sense of T oeplitz forms, ruling out high-entropy pmfs that satisfy r 0 ≥ 1 /q but cannot arise (even approximately) as an autocorrelation. 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