Code spectrum and reliability function: Gaussian channel

Problems of Information T ransmission , v ol. 43, no. 2, pp. 3-24, 2007. M. V. Burnashev CODE SPECTR UM AND RELIABILITY FUNCTION: GA USSIAN CHANNEL 1 A new approa h for upp er b ounding the  hannel reliabilit y funtion using the o de sp etrum is desrib ed. It allo ws to treat b oth lo w and high rate ases in a unied w a y . In partiular, the earlier kno wn upp er b ounds are impro v ed, and a new deriv ation of the sphere-pa king b ound is presen ted.  1. In tro dution and main results W e onsider the disrete time  hannel with indep enden t additiv e Gaussian noise, i.e. if x = ( x 1 , . . . , x n ) is the input o dew ord then the reeiv ed blo  k y = ( y 1 , . . . , y n ) is y i = x i + ξ i , i = 1 , . . . , n , where ( ξ 1 , . . . , ξ n ) are indep enden t Gaussian r.v.'s with E ξ i = 0 , E ξ 2 i = 1 . F or x , y ∈ R n denote ( x , y ) = n P i =1 x i y i , k x k 2 = ( x , x ) , d ( x , y ) = k x − y k 2 and S n − 1 ( b ) = { x ∈ R n : k x k = b } . W e assume that all o dew ords x satisfy the ondition k x k 2 = An , where A > 0 is a giv en onstan t. A subset C = { x 1 , . . . , x M } ⊂ S n − 1 ( √ An ) , M = e Rn , is alled a ( R, A, n ) -o de of rate R and length n . The minimum distan e of the o de C is d ( C ) = min { d ( x i , x j ) : i 6 = j } . The  hannel reliabilit y funtion [1 , 2℄ is dened as E ( R , A ) = lim sup n →∞ 1 n ln 1 P e ( R, A, n ) , where P e ( R, A, n ) is the minimal p ossible deo ding error probabilit y for a ( R, A, n ) -o de. After the fundamen tal results of the pap er [1℄, further impro v emen ts of v arious b ounds for E ( R , A ) ha v e b een obtained in [29℄. In partiular, on the exat form of the funtion E ( R , A ) it w as kno wn only that [ 1℄ E (0 , A ) = A 4 , E ( R , A ) = E sp ( R, A ) , R crit ( A ) ≤ R ≤ C ( A ) , (1) 1 The resear h desrib ed in this publiation w as made p ossible in part b y the Russian F und for F undamen tal Resear h (pro jet n um b er 06-01-00226). 1 where C = C ( A ) = 1 2 ln(1 + A ) , R crit ( A ) = 1 2 ln 2 + A + √ A 2 + 4 4 , (2) E sp ( R, A ) = A 2 − p A (1 − e − 2 R ) g ( R, A ) 2 − ln g ( R, A ) + R , g ( R, A ) = 1 2  p A (1 − e − 2 R ) + p A (1 − e − 2 R ) + 4  . (3) Moreo v er, reen tly [ 8℄ the exat form of E ( R , A ) for a new region R 1 ( A ) ≤ R ≤ R crit ( A ) w as laimed under some restrition on A . Similar to the ase of the binary symmetri  hannel (BSC), that assertion follo ws from a useful observ ation that the tangen t (it has the slop e ( − 1) ) to the funtion E sp ( R, A ) at the p oin t R = R crit ( A ) tou hes the previously kno wn upp er b ound for E ( R , A ) [57℄. Sine those results from [57℄ w ere pro v ed under some restritions on A , those restritions w ere remaining in [8℄ as w ell. Sine there are some inauraies in the form ulation of that result in [8℄ w e do not exp ose orresp onding form ulas from [8℄ (moreo v er, they ha v e a dieren t from ours form). F rom theorem 1 and the form ula ( 9) (see b elo w) the exat form of E ( R , A ) follo ws for the region R 1 ( A ) ≤ R ≤ R crit ( A ) for an y A > 0 . Moreo v er, if A > A 0 ≈ 2 . 2 8 8 (see (14 )) then from theorem 2 b elo w the exat form of E ( R , A ) follo ws for a wider region R 3 ( A ) ≤ R ≤ R crit ( A ) , where R 3 ( A ) < R 1 ( A ) and R 3 ( A ) ≈ R crit ( A ) − 0 . 06866 , A ≥ A 0 . F or 0 < R < R 1 ( A ) , 0 < A ≤ A 0 , or 0 < R < R 3 ( A ) , A > A 0 , still only lo w er and upp er b ounds for E ( R , A ) are kno wn [19℄, and in this pap er the most aurate of the upp er b ounds is impro v ed. W e b egin b y explaining what onstituted the diult y in upp er b ounding the funtion E ( R , A ) in the earlier pap ers [59℄. Note that when testing only t w o o dew ords x i , x j with large distane k x i − x j k 2 = d w e ha v e the deo ding error probabilit y P e ∼ e − d/ 8 . Let B ρn b e the a v erage n um b er of ea h o dew ord x i neigh b ors on the appro ximate distane 2 A (1 − ρ ) n . It w as sho wn in [5℄ that for a ( R, A, n ) -o de there exists ρ su h that B ρn & 2 b ( ρ ) n , where the funtion b ( ρ ) > 0 is desrib ed b elo w, and 2 A (1 − ρ ) n do es not exeed the b est upp er b ound (linear programming) for the minimal o de distane d ( C ) . Therefore, if ea h o dew ord x i has appro ximately B ρn neigh b ors on the distane 2 A (1 − ρ ) n , then it is natural to exp et that P e & B ρn e − A (1 − ρ ) n/ 4 for large n (and not v ery small ρ ), i.e. a v arian t of an additiv e lower b ound for the probabilit y of the union of ev en ts holds. The rst v arian t of su h additiv e b ound w as obtained in [ 5℄ under rather sev ere onstrain ts on R and A . Those results of [5℄ ha v e b een strengthened in [6, 7℄, using the metho d of [1012℄. Ho w ev er there w ere still ertain onstrain ts on R and A . It should b e noted that the in v estigation of E ( R , A ) for the Gaussian  hannel is similar to the in v estigation of E ( R , A ) for the BSC. The dierene is only that due to the disrete struture of a binary alphab et some expressions b eome simpler. F or the BSC the metho d of [6℄ w as reen tly [14, 15 ℄ further dev elop ed. Although the approa h of [14 , 15℄ is still based on [6℄, some additional argumen ts allo w ed the approa h to b e essen tially strengthened and simplied. 2 It should also b e noted that un til the pap ers [14, 15℄, all pap ers men tioned made use of v arious v arian ts of the seond order Bonferroni inequalities. The main aim of this pap er is to pro v e an additiv e b ound without an y onstrain ts on R or A . F or that purp ose the metho d of [14, 15 ℄ is applied. It is also w orth noting that Bonferroni inequalities are not used. This approa h allo ws us to treat b oth lo w and high rate R ases in a unied w a y . As an example, in  2 a new deriv ation of the sphere-pa king b ound is presen ted. In tro due some notations. F or a o de C = { x 1 , . . . , x M } ⊂ S n − 1 ( √ An ) denote ρ ij = ( x i , x j ) An , d ij = k x i − x j k 2 = 2 An (1 − ρ ij ) . (4) Belo w it will b e on v enien t to use the parametri represen tation of the transmission rate R = R ( t ) via the monotoni inreasing funtion R ( t ) = (1 + t ) ln(1 + t ) − t ln t , t ≥ 0 . (5) Consequen tly , for a rate R ≥ 0 in tro due t R ≥ 0 as the unique ro ot of the equation R = R ( t R ) = (1 + t R ) ln(1 + t R ) − t R ln t R . (6) In tro due also the funtions τ ( t ) = 2 p t (1 + t ) 1 + 2 t , τ R = τ ( t R ) . (7) W e shall need the v alues t 1 ( A ) = p 2 + √ 4 + A 2 − 2 4 , τ 1 ( A ) = τ ( t 1 ( A )) = A 2 + √ 4 + A 2 , R 1 ( A ) = R ( t 1 ( A )) , (8) where the funtions τ ( t ) , R ( t ) are dened in ( 7) and (6). Sometimes b elo w w e shall omit the argumen t A in t 1 ( A ) , τ 1 ( A ) , R 1 ( A ) . One of the main results of the pap er is T h e o r e m 1. F or any A > 0 the fol lowing r elations hold: E ( R , A ) =  E sp ( R crit , A ) + R crit − R , R 1 ≤ R ≤ R crit , E sp ( R, A ) , R crit ≤ R ≤ C , (9) and E ( R , A ) ≤ A (1 − τ R ) 4 + ln(1 + 2 t R ) − R , 0 ≤ R ≤ R 1 , (10) wher e R crit ( A ) , R 1 ( A ) , τ R and t R ar e dene d in (2 ), (8), (7) and (6), r esp e tively . R emark 1. W e ha v e R 1 ( A ) < R crit ( A ) , A > 0 . Moreo v er, max A  R crit ( A ) − R 1 ( A )  ≈ 0 . 06866 , and it is attained for A = A 0 ≈ 2 . 2 88 . 3 R emark 2. Note that (see the form ulas (9) and (10 ) for R = R 1 ) E sp ( R crit , A ) + R crit = A (1 − τ 1 ) 4 + ln(1 + 2 t 1 ) . (11) V alidit y of (11) an b e  he k ed using the form ulas (6 ), (7) and the relations 1 + 2 t 1 = r A 4 τ 1 , R crit = 1 2 ln 1 1 − τ 1 , A  1 − e − 2 R crit  = A τ 1 = A τ 1 − 4 , g ( R crit ) = (1 + τ 1 ) √ A 2 √ τ 1 . (12) If A > A 0 ≈ 2 . 2 88 (see (14)) then the upp er b ound ( 10 ) an b e sligh tly impro v ed, and, moreo v er, the v alidit y region of the rst of form ulas (9) an b e enlarged to R 3 ≤ R ≤ R crit , where R 3 ( A ) < R 1 ( A ) (see (14)). T o explain the p ossibilit y of su h an impro v emen t onsider the problem of upp er b ounding the minimal o de distane δ ( R, n ) of a spherial o de. The b est upp er b ound for δ ( R, n ) w as obtained in [4℄ using the linear programming b ound. It w as also notied in [4, p. 20℄ that for R > 0 . 2 3 4 a b etter upp er b ound for δ ( R, n ) is obtained if the linear programming b ound is applied not diretly to the original spherial o de, but to its sub o de on a spherial ap. That observ ation w as reen tly used in [9℄ when estimating the o de sp etrum and the funtion E ( R , A ) . Using the approa h of [ 6℄ an upp er b ound for E ( R , A ) w as obtained in [9℄. But it is rather diult to use that upp er b ound sine it is expressed as an optimization problem o v er four parameters. In fat, it is p ossible to get a more aurate and rather simple b ound that onstitutes theorem 2 b elo w. In tro due the funtion D ( t ) = ln 1 + t t − 1 2 p t (1 + t ) − 1 1 + 2 t , t > 0 , (13) and denote t 2 ≈ 0 . 0 61176 the unique ro ot of the equation D ( t ) = 0 . The equiv alen t equation (with a sign misprin t) app eared earlier in [4, p. 20℄. Denote also R 2 = R ( t 2 ) ≈ 0 . 2339 , τ 2 = τ ( t 2 ) ≈ 0 . 4540 , R 3 ( A ) = R crit ( A ) + R 2 + 1 2 ln(1 − τ 2 ) ≈ R crit ( A ) − 0 . 0687 , A 0 = min  A : R 1 ( A ) ≥ R 2  ≈ 2 . 2 88 . (14) The next result strengthens theorem 1 when A > A 0 . T h e o r e m 2. If A > A 0 ≈ 2 . 288 then the fol lowing r elations hold: E ( R , A ) =  E sp ( R crit , A ) + R crit − R , R 3 ≤ R ≤ R crit , E sp ( R, A ) , R crit ≤ R ≤ C , (15) and E ( R , A ) ≤      1 4 A (1 − τ R ) + ln(1 + 2 t R ) − R , 0 < R ≤ R 2 , 1 4 Aae − 2 R − 1 2 ln(2 − ae − 2 R ) − 1 2 ln a , R 2 ≤ R ≤ R 3 ( A ) , (16) 4 wher e a = (1 − τ 2 ) e 2 R 2 ≈ 0 . 8717 . F or a omparison purp ose w e presen t also the b est kno wn lo w er b ound for the funtion E ( R , A ) [1;3, Theorem 7.4.4℄ E ( R , A ) ≥    A  1 − √ 1 − e − 2 R  / 4 , 0 ≤ R ≤ R low , E sp ( R crit , A ) + R crit − R , R low ≤ R ≤ R crit , E sp ( R, A ) , R crit ≤ R ≤ C ( A ) , (17) where R low ( A ) = 1 2 ln 2 + √ A 2 + 4 4 . (18) Com bining analytial and n umerial metho ds it an b e sho wn that for A > A 0 w e ha v e R low ( A ) < R 2 < R 3 ( A ) < R 1 ( A ) < R crit ( A ) . (19) On the gure the plots of upp er (15 ),(16) and lo w er ( 17 ) b ounds for E ( R , A ) with A = 4 are presen ted. The pap er is organized as follo ws. In 2 the main analytial to ol (prop osition 1) is presen ted and, as an example, the sphere-pa king upp er b ound is deriv ed. In 3 prop osition 1 and the o de sp etrum are om bined in prop ositions 23. In 4 (using results of 3 and the kno wn b ound for the o de sp etrum - theorem 3) theorem 1 is pro v ed. In 5 theorem 2 is pro v ed. Pro ofs of some auxiliary results are presen ted in App endix.  2. New approa h and sphere-pa king exp onen t F or the onditional output probabilit y distribution densit y p ( y | x ) of the input o dew ord x the form ula holds ln p ( y | x ) = − 1 2 d ( y , x ) − n 2 ln(2 π ) , x , y ∈ R n (in a similar form ula in [6 ℄ there is a misprin t - the min us sign is missing). T o desrib e our approa h, w e x a small δ = o (1) , n → ∞ , and s > 0 and for an output y dene the set: X s ( y ) = { x i ∈ C : | d ( y , x i ) − sn | ≤ δ n } , y ∈ R n . (20) All o dew ords { x i } are assumed equiprobable. F or a  hosen deo ding metho d denote P ( e | y , x i ) the onditional deo ding error probabilit y pro vided that x i w as transmitted and y w as reeiv ed. Denote p e ( y ) the probabilit y distribution densit y to get the output y and to mak e a deo ding error. Then p e ( y ) = M − 1 M X i =1 p ( y | x i ) P ( e | y , x i ) ≥ M − 1 X x i ∈ X s ( y ) p ( y | x i ) P ( e | y , x i ) = = M − 1 (2 π ) − n/ 2 X x i ∈ X s ( y ) e − d ( y , x i ) / 2 P ( e | y , x i ) ≥ ≥ M − 1 (2 π e s + δ ) − n/ 2 X x i ∈ X s ( y ) P ( e | y , x i ) ≥ M − 1 (2 π e s + δ ) − n/ 2 [ | X s ( y ) | − 1] + , 5 where [ z ] + = max { 0 , z } and | A |  the ardinalit y of the set A . F or the deo ding error probabilit y P e w e get P e = Z y ∈ R n p e ( y ) d y ≥ M − 1 (2 π e s + δ ) − n/ 2 Z y : | X s ( y ) | ≥ 2 [ | X s ( y ) | − 1] d y . Sine ( a − 1) ≥ a/ 2 , a ≥ 2 , w e ha v e P e ≥ (2 M ) − 1 (2 π e s + δ ) − n/ 2 Z y : | X s ( y ) | ≥ 2 | X s ( y ) | d y , (21) where X s ( y ) is dened in ( 20 ). T o dev elop further the righ t-hand side of ( 21) w e x some r > 0 and for ea h x i in tro due the set Z s,r ( i ) =  y :   k y k 2 − r n   ≤ δ n , | d ( y , x i ) − sn | ≤ δ n, | X s ( y ) | ≥ 2  = =  y : |k y k 2 − r n | ≤ δ n , | d ( y , x i ) − sn | ≤ δ n and there exists x j 6 = x i with | d ( x j , y ) − sn | ≤ δ n  . (22) F or a measurable set A ⊆ R n denote b y m ( A ) its Leb esque measure. Then Z y : | X s ( y ) | ≥ 2 | X s ( y ) | d y ≥ M X i =1 m ( Z s,r ( i )) and from (21 ) w e get P r o p o s i t i o n 1. With any δ > 0 for the de  o ding err or pr ob ability P e the lower b ound holds P e ≥ 1 2 M max s,r ( (2 π e s + δ ) − n/ 2 M X i =1 m ( Z s,r ( i )) ) , (23) wher e Z s,r ( i ) is dene d in ( 22 ). Example: sphere-pa king upp er b ound. W e sho w rst ho w to get the sphere- pa king upp er b ound E ( R , A ) ≤ E sp ( R, A ) from (23) (f. [1;3, Chapter 7.4℄). T o simplify form ulas w e write b elo w a ≈ b if | a − b | ≤ δ , where δ = o (1) , n → ∞ . Note that Z s,r ( i ) = Z (1) s,r ( i ) \ Z (2) s,r ( i ) , Z (1) s,r ( i ) =  y : k y k 2 /n ≈ r , d ( y , x i ) /n ≈ s  , Z (2) s,r ( i ) =  y : k y k 2 /n ≈ r , d ( y , x i ) /n ≈ s , | X s ( y ) | = 1  = =  y : k y k 2 /n ≈ r , d ( y , x i ) /n ≈ s and there is no x j 6 = x i with d ( x j , y ) /n ≈ s  . Then w e ha v e M [ i =1 Z (2) s,r ( i ) = Y s =  y : k y k 2 /n ≈ r , | X s ( y ) | = 1  = =  y : k y k 2 /n ≈ r and there exists exatly one x i with d ( y , x i ) /n ≈ s  , Y s ⊆ Y ( r ) =  y : k y k 2 /n ≈ r  , 6 and the lo w er b ound (23 ) tak es the form P e ≥ (2 M ) − 1 (2 π e s + δ ) − n/ 2 h M m  | Z (1) s,r (1) |  − m ( Y ( r )) i + . The surfae area of a n -dimensional sphere of radius a is S n ( a ) = nπ n/ 2 a n − 1 / Γ( n/ 2 + 1) ∼ (2 π ea 2 /n ) n/ 2 . Then from a standard geometry w e get m  | Z (1) s,r (1) |  ∼ (2 πer 1 ) n/ 2 , m ( Y ( r )) ∼ (2 π er ) n/ 2 , r 1 = s − ( r − A − s ) 2 4 A = r − ( r + A − s ) 2 4 A . Therefore the lo w er b ound (23 ) tak es the form P e & M − 1 ( e s + δ − 1 ) − n/ 2 h M r n/ 2 1 − r n/ 2 i + . (24) W e w an t to maximize the righ t-hand side of (24) o v er s, r . Sine w e are in terested only in exp onen ts in n , w e ma y assume that M r n/ 2 1 = r n/ 2 , i.e. e 2 R r 1 = r . Then w e should maximize the funtion f ( s, r ) = ln r − s pro vided s − ( r − A − s ) 2 4 A − r e − 2 R = 0 . As usual, onsidering the funtion g ( s, r ) = ln r − s + λ  s − ( r − A − s ) 2 4 A − r e − 2 R  , and solving the equations g ′ s = g ′ r = 0 , w e get r = 1 1 − λ (1 − e − 2 R ) , s = r + A − 2 A λ , where λ satises the equation  1 − e − 2 R  λ 2 + A  1 − e − 2 R  λ − A = 0 . Therefore λ = √ A g 1 √ 1 − e − 2 R , where g 1 = g 1 ( R, A ) is dened in ( 3). Note that g 2 − 1 = g p A (1 − e − 2 R ) , 1 − λ  1 − e − 2 R  = 1 g 2 , ln r − s = 2 ln g − 1 − A + g p A (1 − e − 2 R ) . 7 T aking in to aoun t that e 2 R r 1 = r , w e get from (24 ) and (3) 1 n ln 1 P e ≤ s − 1 2 − ln r 1 = s − 1 2 + R − 1 2 ln r = = A − p A (1 − e − 2 R ) g ( R, A ) 2 − ln g ( R, A ) + R = E sp ( R, A ) , whi h giv es the sphere-pa king upp er b ound E ( R , A ) ≤ E sp ( R, A ) .  3. Lo w er b ound (23) and o de sp etrum F or a o de C ⊂ S n − 1 ( √ An ) in tro due the o de sp etrum funtion B ( s, t ) = 1 |C |      u , v ∈ C : s ≤ ( u , v ) An < t      , (25) and denote b ( ρ, ε ) = 1 n ln B ( ρ − ε, ρ + ε ) , 0 < ε < ρ . T o simplify notation w e write b elo w a ≈ b if | a − b | ≤ δ , where δ = 1 / √ An . F or some r > 0 w e onsider only the set of outputs Y ( r ) =  y : k y k 2 /n ≈ r  ⊆ R n . (26) T o in v estigate the funtion E ( R , A ) , R < R crit , w e use a v arian t of the lo w er b ound (23 ) P e ≥ (2 M ) − 1 max s,r > 0 max ρ ( (2 π e s + δ ) − n/ 2 M X i =1 m ( Z s,r ( ρ, i )) ) , (27) where Z s,r ( ρ, i ) =  y ∈ Y ( r ) : there exists x j with ρ ij ≈ ρ and d ( x i , y ) /n ≈ d ( x j , y ) /n ≈ s  , (28) and ρ ij is dened in (4). W e dev elop the lo w er b ound (27 ), relating it to the o de sp etrum (25 ), i.e. to the distribution of the pairwise inner pro duts { ρ ij } . F or o dew ords x i , x j with ρ ij ≈ ρ in tro due the set Z s,r ( ρ, i, j ) =  y ∈ Y ( r ) : d ( x i , y ) /n ≈ d ( x j , y ) /n ≈ s  . (29) Then for an y i from (28 ) and (29) w e ha v e Z s,r ( ρ, i ) = [ j : ρ ij ≈ ρ Z s,r ( ρ, i, j ) . (30) Denoting Z ( s, r, ρ ) = m ( Z s,r ( ρ, i, j )) (31) 8 (sine the measure of that set do es not dep end on indies ( i, j ) ), w e ha v e (see App endix) 1 n ln Z ( s, r, ρ ) = 1 2 ln [2 π ez ( s, r , ρ )] + o (1) , n → ∞ , (32) where z ( s, r , ρ ) = r − ( A + r − s ) 2 2 A (1 + ρ ) . (33) Note that due to (30 ), for the sum in the righ t-hand side of (27 ) for an y ρ w e ha v e M X i =1 m ( Z s,r ( ρ, i )) ≤ X ( i,j ): ρ ij ≈ ρ m ( Z s,r ( ρ, i, j )) = Z ( s, r, ρ ) |{ ( i, j ) : ρ ij ≈ ρ }| = = exp n n 2 ln [2 π ez ( s, r , ρ )] + [ R + b ( ρ )] n + o ( n ) o , (34) sine for b ( ρ ) = b ( ρ, δ ) the follo wing form ula holds (see (25 )) |{ ( i, j ) : ρ ij ≈ ρ }| = e Rn B ( ρ − δ, ρ + δ ) = e ( R + b ( ρ )) n . Supp ose that for some ρ = ρ 0 in the relation ( 34 ) the follo wing asymptoti equalit y holds: 1 n ln " M X i =1 m ( Z s,r ( ρ 0 , i )) # = 1 2 ln [2 π ez ( s, r , ρ 0 )] + R + b ( ρ 0 ) + o (1) , n → ∞ . (35) Using the funtions s = s ( ρ ) , r = r ( ρ ) (they are  hosen b elo w), from (27), (35 ) and (33 ) for su h ρ 0 w e get 1 n ln 1 P e ≤ s − 1 2 − 1 2 ln  r − ( A + r − s ) 2 2 A (1 + ρ 0 )  − b ( ρ 0 ) + o (1) . (36) W e set b elo w s ( ρ ) = A (1 − ρ ) 2 + 1 , r ( ρ ) = A (1 + ρ ) 2 + 1 . (37) Su h  hoie of s ( ρ ) , r ( ρ ) minimizes (o v er s, r ) the righ t-hand side of ( 36). Optimalit y of su h s, r an also b e dedued from the form ulas (72 ) (see App endix). F or su h s ( ρ ) , r ( ρ ) w e ha v e r − ( A + r − s ) 2 / [2 A (1 + ρ )] = 1 , and then ( 36 ) tak es the simple form 1 n ln 1 P e ≤ A (1 − ρ 0 ) 4 − b ( ρ 0 ) + o (1) . (38) Note that b ( ρ ) ≥ 0 if there exists a pair ( x i , x j ) with ρ ij ≈ ρ , and b ( ρ ) = −∞ if there is no an y pair with ρ ij ≈ ρ . W e form ulate the result obtained as follo ws. P r o p o s i t i o n 2. If for some ρ 0 the  ondition (35) is full le d, then the ine quality (38 ) for the de  o ding err or pr ob ability P e holds . 9 W e sho w that as su h ρ 0 w e ma y  ho ose the v alue ρ 0 , minimizing the righ t-hand side of (38). In other w ords, dene ρ 0 as follo ws Aρ 0 + 4 b ( ρ 0 ) = max | ρ |≤ 1 { Aρ + 4 b ( ρ ) } . (39) R emark 3. If there are sev eral su h ρ 0 , w e ma y use an y of them. It is not imp ortan t that w e do not kno w the funtion b ( ρ ) . W e ma y use as b ( ρ ) an y lo w er b ound for it (see pro ofs of theorems 1 and 2). P r o p o s i t i o n 3. F or ρ 0 fr om (39) the  ondition (35) holds and ther efor e the ine quality (38) is valid . P r o o f. It is on v enien t to quan tize the range of p ossible v alues of the normalized inner pro duts ρ ij . F or that purp ose w e partition the whole range [ − 1; 1] of v alues ρ ij on subin terv als of the length δ = 1 / √ An . There will b e n 1 = 2 /δ of su h subin terv als. W e ma y assume that ρ ij tak es v alues from the set {− 1 = ρ 1 < . . . < ρ n 1 = 1 } . W e all ( x i , x j ) a ρ -pair if ( x i , x j ) / ( An ) ≈ ρ . Then M e nb ( ρ ) is the total n um b er of ρ -pairs. W e use s = s ( ρ 0 ) , r = r ( ρ 0 ) from (37 ) and onsider only outputs y ∈ Y ( r ) = Y ( r ( ρ 0 )) . W e sa y that su h a p oin t y is ρ -o v ered if there exists a ρ -pair ( x i , x j ) su h that d ( x i , y ) /n ≈ d ( x j , y ) /n ≈ s . Then the total (taking in to aoun t the o v ering m ultipliities) Leb esque measure of all ρ -o v ered p oin ts y equals M e nb ( ρ ) Z ( s, r, ρ ) . In tro due the set Y ( ρ 0 , ρ ) of all ρ -o v ered p oin ts y Y ( ρ 0 , ρ ) = { y ∈ Y ( r ) : y is ρ − o v ered } . W e onsider the set Y ( ρ 0 , ρ ) and p erform its leaning, exluding from it all p oin ts y that are also ρ -o v ered for an y ρ su h that | ρ − ρ 0 | ≥ 4 δ , i.e. w e onsider the set Y ′ ( ρ 0 , ρ 0 ) = Y ( ρ 0 , ρ 0 ) \ [ | ρ − ρ 0 |≥ 4 δ Y ( ρ 0 , ρ ) = =  y ∈ Y ( r ) : y is ρ 0 − o v ered and is not ρ − o v ered for an y ρ su h that | ρ − ρ 0 | ≥ 4 δ  . (40) Ea h p oin t y ∈ Y ′ ( ρ 0 , ρ 0 ) an b e ρ -o v ered only if | ρ − ρ 0 | < 4 δ . W e sho w that b oth sets Y ( ρ 0 , ρ 0 ) and Y ′ ( ρ 0 , ρ 0 ) ha v e essen tially the same Leb esque measures. Note that a ρ -pair ( x i , x j ) ρ -o v ers the set Z s,r ( ρ, i, j ) from (29) with the Leb esque measure Z ( s, r, ρ ) . W e ompare the v alues P | ρ − ρ 0 |≥ 4 δ e nb ( ρ ) Z ( s, r, ρ ) and e nb ( ρ 0 ) Z ( s, r, ρ 0 ) (see (40)). F or that purp ose w e onsider the funtion g ( ρ ) = 1 n ln e nb ( ρ ) Z ( s, r, ρ ) e nb ( ρ 0 ) Z ( s, r, ρ 0 ) = b ( ρ ) − b ( ρ 0 ) + 1 2 ln z ( s, r , ρ ) z ( s, r , ρ 0 ) + o (1) , (41) where z ( s, r , ρ ) is dened in (33 ). F rom (33 ) w e also ha v e z ( s, r , ρ ) = 1 + A (1 + ρ 0 )( ρ − ρ 0 ) 2(1 + ρ ) . 10 Sine b ( ρ ) ≤ b ( ρ 0 ) − A ( ρ − ρ 0 ) / 4 (see (39)), for the funtion g ( ρ ) from (41 ) w e get g ( ρ ) ≤ 1 2 ln  1 + A (1 + ρ 0 )( ρ − ρ 0 ) 2(1 + ρ )  − A ( ρ − ρ 0 ) 4 ≤ − A ( ρ − ρ 0 ) 2 4(1 + ρ ) . (42) Sine ρ − ρ 0 = iδ , | i | ≥ 4 , after simple alulations w e ha v e P | ρ − ρ 0 |≥ 4 δ e nb ( ρ ) Z ( s, r, ρ ) e nb ( ρ 0 ) Z ( s, r, ρ 0 ) = X | ρ − ρ 0 |≥ 4 δ e ng ( ρ ) ≤ 2 X i ≥ 4 exp  − Anδ 2 i 2 8  = 2 X i ≥ 4 e − i 2 / 8 < 1 2 . Therefore w e get e nb ( ρ 0 ) Z ( s, r, ρ 0 ) − X | ρ − ρ 0 |≥ 4 δ e nb ( ρ ) Z ( s, r, ρ ) > 1 2 e nb ( ρ 0 ) Z ( s, r, ρ 0 ) . Then the total (taking in to aoun t the o v ering m ultipliities) Leb esque measure of all ρ -o v ered p oin ts y ∈ Y ′ ( ρ 0 , ρ 0 ) exeeds M e nb ( ρ 0 ) Z ( s, r, ρ 0 ) / 2 . Remind that an y p oin t y ∈ Y ′ ( ρ 0 , ρ 0 ) an b e ρ -o v ered only if | ρ − ρ 0 | < 4 δ . F or ea h p oin t y ∈ Y ′ ( ρ 0 , ρ 0 ) onsider the set X s ( y ) dened in ( 20 ), i.e. the set of all o dew ords { x i } su h that d ( x i , y ) /n ≈ s . The o dew ords from X s ( y ) satisfy also the ondition | ( x i , x j ) / ( An ) − ρ 0 | < 4 δ , i.e. the set { x i } onstitutes almost a simplex. It is rather lear that the n um b er | X s ( y ) | of su h o dew ords is not exp onen tial on n , i.e. max y ∈ Y ′ ( ρ 0 ,ρ 0 )  1 n ln | X s ( y ) |  = o (1) , n → ∞ . (43) F ormally the v alidit y of (43 ) follo ws from lemma 2 (see b elo w). Note that if A 1 , . . . , A N ⊂ R n are a measurable sets, and an y p oin t a ∈ S i A i is o v ered b y the sets { A i } not more than K times, then m N [ i =1 A i ! ≥ 1 K N X i =1 m ( A i ) . (44) F or y ∈ Y ′ ( ρ 0 , ρ 0 ) denote X i ( y ) =  x j : d ( x i , y ) /n ≈ d ( x j , y ) /n ≈ s, ρ ij ≈ ρ 0  , X max = max i, y ∈ Y ′ ( ρ 0 ,ρ 0 ) | X i ( y ) | . (45) Due to (43 ) w e ha v e 1 n ln X max = o (1) , n → ∞ . (46) 11 Sine an y p oin t y ∈ Y ′ ( ρ 0 , ρ 0 ) an b e ρ -o v ered not more than X max times and Y ′ ( ρ 0 , ρ 0 ) ⊆ Y ( ρ 0 , ρ 0 ) , then from (43 )(46 ) w e get 1 n ln " M X i =1 m ( Z s,r ( ρ 0 , i )) # ≥ 1 n ln m ( Y ′ ( ρ 0 , ρ 0 )) ≥ ≥ 1 n ln  M e nb ( ρ 0 ) Z ( s, r, ρ 0 )  + o (1) = = 1 2 ln [2 π ez ( s, r , ρ 0 )] + R + b ( ρ 0 ) + o (1) , n → ∞ . (47) Therefore due to the inequalities (34 ) and (47), the ondition (35 ) is fullled, and then the relation (38 ) holds. T o omplete the pro of of prop osition 2 it remains to establish the form ula (43 ). W e pro v e it rst for a simpler (but a more natural) ase ρ ∗ ≤ τ 1 , and then onsider the general ase. C a s e ρ 0 ≤ τ 1 . In that ase the relation (43) follo ws from simple lemma (see pro of in App endix). L e m m a 1. L et y ∈ R n with k y k 2 = r n . L et C = { x 1 , . . . , x M } ⊂ S n − 1 ( √ An ) b e a  o de with k x i − y k 2 = sn, i = 1 , . . . , M , and max i 6 = j ( x i , x j ) ≤ Anρ . If A + r − s ≥ 2 p Ar ρ , (48) then M ≤ 2 n . F or s ( ρ ) , r ( ρ ) from (37 ) the ondition ( 48) holds, if ρ ≤ A 2 + √ 4 + A 2 = τ 1 ( A ) . (49) F rom lemma 1 and (49) the relation (43 ) follo ws. G e n e r a l  a s e. Although a o de with ρ 0 > τ 1 an hardly derease the deo ding error probabilit y P e , its in v estigation needs a bit more eorts. The relation ( 43) follo ws from lemma (see pro of in App endix). L e m m a 2. L et for a  o de C = { x 1 , . . . , x M } ⊂ S n − 1 ( √ An ) and some ρ < 1 it holds that max i 6 = j | ( x i , x j ) − Aρn | = o ( n ) , n → ∞ . Then ln M = o ( n ) , n → ∞ . It ompletes the pro of of prop osition 3. N Using prop osition 3 and t w o lo w er b ounds for b ( ρ ) w e shall pro v e theorems 1 and 2.  4. Pro of of theorem 1 12 First w e in v estigate the funtion E ( R , A ) for 0 < R ≤ R 1 ( A ) and pro v e the upp er b ound (10 ). Then for R 1 ( A ) < R < R crit ( A ) , using the straigh t-line b ound [2 ℄, w e will pro v e the form ula (9). T o apply prop osition 3 w e use the kno wn b ound for the o de sp etrum. The next result is a sligh t renemen t of [5, Theorem 9℄ (see also [6, Theorem 1℄). T h e o r e m 3. L et C ⊂ S n − 1 ( √ An ) b e a  o de with |C | = e Rn , R > 0 . Then for any ε = ε ( n ) > 0 ther e exists ρ suh that ρ ≥ τ R and b ( ρ ) = 1 n ln B ( ρ − ε, ρ + ε ) ≥ R − J ( t R , ρ ) + ln ε n + o (1) , n → ∞ , J ( t, ρ ) = (1 + 2 t ) ln [2 tρ + q ( t, ρ )] − ln q ( t, ρ ) − t ln[4 t (1 + t )] , q ( t, ρ ) = ρ + p (1 + 2 t ) 2 ρ 2 − 4 t (1 + t ) , (50) wher e t R , τ R ar e dene d in (4) and (7), and o (1) do es not dep end on ε . Note that J ′ ρ ( t, ρ ) = 4 t (1 + t ) ρ + p (1 + 2 t ) 2 ρ 2 − 4 t (1 + t ) , J ′′ ρρ ( t, ρ ) = − 4 t (1 + t ) [ ρ + p (1 + 2 t ) 2 ρ 2 − 4 t (1 + t )] 2 " 1 + (1 + 2 t ) 2 ρ p (1 + 2 t ) 2 ρ 2 − 4 t (1 + t ) # , J ′ t ( t, ρ ) = 2 ln [2 tρ + q ( t, ρ )] − ln[4 t (1 + t )] , [ R ( t ) − J ( t, ρ )] ′ t = 2 ln 2(1 + t ) 2 tρ + q > 0 , J ( t R , τ R ) = ln(1 + 2 t R ) , J ( t R , 1) = R . (51) P r o p o s i t i o n 4. F or the funtion E ( R , A ) the upp er b ound (10) holds . P r o o f. Due to theorem 2 there exists ρ ≥ τ R su h that the inequalit y (50 ) holds. Denote ρ ∗ the largest of su h ρ . Sine b ( ρ 0 ) ≥ b ( ρ ∗ ) − A ( ρ 0 − ρ ∗ ) / 4 (ñì. ( 39 )), from (38 ) and (50 ) w e get 1 n ln 1 P e ≤ A (1 − ρ 0 ) 4 − b ( ρ 0 ) + o (1) ≤ A (1 − ρ ∗ ) 4 − b ( ρ ∗ ) + o (1) ≤ ≤ A (1 − ρ ∗ ) 4 + J ( t R , ρ ∗ ) − R + o (1) . (52) Note that if τ R ≤ τ 1 (i.e. if R ≤ R 1 ( A ) ) then (see App endix) [ J ( t R , ρ ) − Aρ/ 4] ′ ρ ≤ 0 , ρ ≥ τ R , (53) and therefore the funtion J ( t R , ρ ) − Aρ/ 4 monotone dereases on ρ ≥ τ R . Sine ρ ∗ ≥ τ R then for τ R ≤ τ 1 w e an on tin ue (52 ) as follo ws 1 n ln 1 P e ≤ A (1 − τ R ) 4 + J ( t R , τ R ) − R + o (1) = = A (1 − τ R ) 4 + ln(1 + 2 t R ) − R , 0 < R ≤ R 1 , (54) 13 whi h is the desired upp er b ound (10 ). N T o pro v e the relation (9 ) note that the b est upp er b ound for E ( R , A ) is a om bination of the upp er b ound (10) and the sphere-pa king b ound via the straigh t-line b ound [ 2℄, whi h giv es E ( R , A ) ≤ A (1 − τ 1 ) 4 + ln(1 + 2 t 1 ) − R , R 1 ≤ R ≤ R crit . On the other hand, the random o ding b ound [ 1, 3℄ giv es E ( R , A ) ≥ E sp ( R crit , A ) + R crit − R , R ≤ R crit , where E sp ( R, A ) is dened in ( 3 ). T ogether with the form ula (11 ) it ompletes the pro of of theorem 1. N  5. Pro of of theorem 2 As w as already men tioned in  1, for R > 0 . 234 the upp er b ounds for the minimal o de distane [4, p. 20℄ of a spherial o de and its sp etrum [9℄ an b e impro v ed, if the linear programming b ound is not diretly applied to the original spherial o de, but to its sub o des on spherial aps. The same approa h allo ws to impro v e the upp er b ound for E ( R , A ) as w ell. F or that purp ose w e will need a b ound for a o de sp etrum b etter than (50 ). The b ound obtained b elo w (theorem 4), probably , is equiv alen t to the similar b ound in [9, Theorem 3℄ (expressed in a dieren t terms), but its deriv ation is simpler and a more aurate. Sine w e are in terested only in angles b et w een o dew ords x i , x j , for the form ulas simpliation w e ma y set An = 1 , and onsider a o de C ⊂ S n − 1 (1) = S n − 1 . Let T n θ ( z ) b e the spheri al  ap with half-angle 0 ≤ θ ≤ π / 2 and en ter z ∈ S n − 1 , i.e. T n θ ( z ) =  x ∈ S n − 1 : ( x , z ) ≥ cos θ  . It will b e on v enien t to onsider sub o des of C not on spherial aps T n θ ( z ) , but on related with them thin ring-shap ed surfaes D n θ ( z ) . W e set further δ = 1 /n 2 , and denote D n θ ( z ) as D n θ ( z ) = T n θ ( z ) \ T n θ − δ ( z ) =  x ∈ S n − 1 : cos θ ≤ ( x , z ) ≤ cos( θ − δ )  . (55) Denote D n ( θ ) the surfae area of D n θ ( z ) . Then [ 1, form ula (21)℄ D n ( θ ) = ( n − 1) π ( n − 1) / 2 Γ(( n + 1) / 2) θ Z θ − δ sin n − 2 u du , δ ≤ θ ≤ π / 2 . It is not diult to sho w that 1 − 1 2 n sin θ ≤ D n ( θ )Γ(( n + 1) / 2) n 2 π ( n − 1) / 2 ( n − 1) sin n − 2 θ ≤ 1 . 14 Sine the surfae area | S n − 1 | of the sphere S n − 1 equals nπ n/ 2 / Γ( n/ 2 + 1) , w e ha v e uniformly o v er 1 /n ≤ θ ≤ π / 2 1 n ln D n ( θ ) | S n − 1 | = ln sin θ + o (1) , n → ∞ . F or the o de C ⊂ S n − 1 and θ su h that max { arcsin e − R , 1 /n } ≤ θ ≤ π / 2 , and z ∈ S n − 1 w e onsider the sub o de C ( θ , z ) = C ∩ D n θ ( z ) with |C ( θ , z ) | = e nr ( z ) o dew ords. Then 1 m ( S n − 1 ) Z z ∈ S n − 1 |C ( θ , z ) | d z = |C | D n ( θ ) | S n − 1 | = exp { ( R + ln sin θ ) n + o ( n ) } , i.e. in a v erage (o v er z ∈ S n − 1 ) a sub o de C ( θ , z ) has the rate r = R + ln sin θ + o (1) . All its |C ( θ , z ) | o dew ords are lo ated in the ball B n (sin θ , z ′ ) of radius sin θ and en tered at z ′ = z cos θ . Moreo v er, they are lo ated in a thin (of thi kness ∼ δ ) torus orthogonal to z . If x ∈ D n θ ( z ) , then w e denote x ′ = x − z ′ the orresp onding v etor from B n (sin θ , z ′ ) . The original angle ϕ b et w een t w o v etors x , y ∈ D n θ ( z ) b eomes the angle ϕ ′ + O ( δ ) b et w een the v etors x ′ , y ′ ∈ B n (sin θ , z ′ ) , where sin( ϕ ′ / 2) = sin ( ϕ/ 2) / sin θ . The original v alue ρ = c os ϕ b eomes the v alue ρ ′ + O ( δ ) , where ρ ′ = cos ϕ ′ is dened b y the form ula 1 − ρ = (1 − ρ ′ ) sin 2 θ , (56) sine ρ ′ = cos  2 arcsin  sin( ϕ/ 2) sin θ  = 1 − 2 sin 2 ( ϕ/ 2) sin 2 θ = 1 − (1 − ρ ) e 2( R − r ) . The angle ϕ ′ and the v alue ρ ′ orresp ond to the ase when the v etors x ′ , y ′ are orthogonal to z . The o de C ( θ , z ) is then transferred to the o de C ′ ( z ) = C ′ ( θ , z ) ⊂ B n (sin θ , z ′ ) . T o ev aluate the a v erage n um b er e nb C ( ρ ) of ρ -neigh b ors in the o de C , w e onsider an y pair x i , x j with ( x i , x j ) = ρ and in tro due the sets Z ( x , a ) =  z ∈ S n − 1 : ( x , z ) ≥ a  , Z ( x , y , a ) =  z ∈ S n − 1 : ( x , z ) ≥ a and ( y , z ) ≥ a  . Denote b y Ω n ( θ ) the surfae area of the spherial ap T n θ ( z ) . F or 0 ≤ θ < π / 2 w e ha v e Ω n ( θ ) = π ( n − 1) / 2 sin n − 1 θ Γ(( n + 1) / 2) cos θ (1 + o (1)) , n → ∞ . Then for the Leb esque measure m ( a ) of the set Z ( x , a ) w e ha v e m ( a ) = m ( Z ( x , a )) = Ω n (arccos a ) . 15 W e ev aluate the Leb esque measure m ( ρ, a ) of the set Z ( x , y , a ) pro vided ( x , y ) = ρ . Note that if x , y ∈ S n − 1 and ( x , y ) = ρ , then k x + y k 2 = 2(1 + ρ ) . Therefore v = ( x + y ) / p 2(1 + ρ ) ∈ S n − 1 , and then Z ( x , y , a ) ⊆  z ∈ S n − 1 : ( x + y , z ) ≥ 2 a  = = n z ∈ S n − 1 : ( v , z ) ≥ a p 2 / (1 + ρ ) o = Z  v , a p 2 / (1 + ρ )  . Therefore w e get m ( ρ, a ) = m ( Z ( x , y , a )) ≤ m  Z  v , a p 2 / (1 + ρ )  = Ω n  arccos  a p 2 / (1 + ρ )  . That upp er b ound for m ( ρ, a ) is logarithmially (as n → ∞ ) exat. In partiular, if a = cos θ and ( x , y ) = ρ , then 1 n ln m (cos θ ) m ( ρ, cos θ ) ≥ ln sin θ − ln sin  arccos  p 2 / (1 + ρ ) cos θ  = = ln sin θ − ln p 1 − 2 cos 2 θ / (1 + ρ ) . W e use b elo w the v alues ρ ′ = ρ ′ ( ρ, θ ) from and ( 56 ) and ε ′ = ε/ sin 2 θ . Then denoting B C ( ρ ) = B C ( ρ − ε, ρ + ε ) , B C ′ ( z ) ( ρ ′ ) = B C ′ ( z ) ( ρ ′ − ε ′ , ρ ′ + ε ′ ) , for an y ρ, ε w e ha v e B C ( ρ ) |C | = 1 m ( ρ, cos θ ) Z z ∈ S n − 1 B C ′ ( z ) ( ρ ′ ) |C ′ ( z ) | d z . (57) Indeed, the v alue B C ( ρ ) |C | is the total n um b er of pairs x i , x j ∈ C with | ( x i , x j ) − ρ | ≤ ε , and B C ′ ( z ) ( ρ ′ ) |C ′ ( z ) | is the total n um b er of similar pairs x ′ i , x ′ j ∈ C ′ ( z ) with | ( x ′ i , x ′ j ) / ( k x ′ i k · k x ′ j k ) − ρ ′ | ≤ ε ′ . Moreo v er, ea h pair x ′ i , x ′ j ∈ C ′ ( z ) giv es the on tribution m ( ρ, cos θ ) to the in tegral, from whi h the form ula (57) follo ws. F rom (57 ) for an y set A ⊆ S n − 1 w e ha v e e nb C ( ρ ) ≥ 1 m ( ρ, cos θ ) |C | Z z ∈A e nb C ′ ( z ) ( ρ ′ ) |C ′ ( z ) | d z , (58) and also |C | = 1 m (cos θ ) Z z ∈ S n − 1 |C ′ ( z ) | d z ≥ 1 m (cos θ ) Z z ∈A |C ′ ( z ) | d z . The o de C ′ ( z ) has the rate r ( z ) = (ln |C ′ ( z ) | ) /n . Then there exists r 0 su h that |C | = e o ( n ) m (cos θ ) max t  e tn m  z ∈ S n − 1 : | r ( z ) − t | ≤ ε  = e r 0 n + o ( n ) m ( S 0 ) m (cos θ ) , S 0 =  z ∈ S n − 1 : | r ( z ) − r 0 | ≤ ε  . (59) Sine m ( S 0 ) ≤ m ( S n − 1 ) then r 0 ≥ 1 n ln |C | m (cos θ ) m ( S n − 1 ) = R + ln sin θ + o (1) . (60) 16 W e set A = S 0 and ε = o (1) , n → ∞ . Then using the Jensen inequalit y , from ( 58 ) and (59 ) w e ha v e e nb C ( ρ ) ≥ 1 m ( ρ, cos θ ) |C | Z z ∈ S 0 e nb C ′ ( z ) ( ρ ′ ) |C ′ ( z ) | d z ≥ ≥ m (cos θ ) e o ( n ) m ( ρ, cos θ ) m ( S 0 ) Z z ∈ S 0 e nb C ′ ( z ) ( ρ ′ ) d z ≥ ≥ m (cos θ ) e o ( n ) m ( ρ, cos θ ) exp    n m ( S 0 ) Z z ∈ S 0 b C ′ ( z ) ( ρ ′ ) d z    , from whi h w e get b C ( ρ ) ≥ 1 n ln m (cos θ ) m ( ρ, cos θ ) + 1 m ( S 0 ) Z z ∈ S 0 b C ′ ( z ) ( ρ ′ ) d z + o (1) . (61) Due to theorem 3 for ea h o de C ′ ( z ) , z ∈ S 0 , there exists ρ ′′ = ρ ′′ ( z ) su h that ρ ′′ ≥ τ r 0 and b C ′ ( z ) ( ρ ′′ ) ≥ r 0 − J ( t r 0 , ρ ′′ ) + o (1) . Therefore there exists ρ ′ ≥ τ r 0 and the orresp onding ρ = ρ ( ρ ′ ) from (56) su h that from the inequalit y (61) w e get b C ( ρ ) ≥ 1 n ln m (cos θ ) m ( ρ, cos θ ) + r 0 − J ( t r 0 , ρ ′ ) + o (1) ≥ = 1 n ln m (cos θ ) m ( ρ, cos θ ) + R + ln sin θ − J ( t R +ln sin θ , ρ ′ ) + o (1) ≥ ≥ R + 2 ln sin θ − J ( t R +ln si n θ , ρ ′ ) − ln p 1 − 2 cos 2 θ / (1 + ρ ) + o (1) = = R + ln sin θ − J ( t R +ln sin θ , ρ ′ ) + 1 2 ln (1 + ρ ) (1 + ρ ′ ) + o (1) , (62) where w e used the form ula (60) and monotoniit y of the funtion r − J ( t r , ρ ) on r (see (51 )), and ρ ′ = ρ ′ ( ρ, θ ) is dened in ( 56 ). After the v ariable  hange sin θ = e r − R from (62 ) w e get T h e o r e m 4. L et C ⊂ S n − 1 (1) b e a  o de with |C | = e Rn , R > 0 . Then for any r ≤ R ther e exists ρ ′ suh that ρ ′ ≥ τ r and for ρ = 1 − (1 − ρ ′ ) e 2( r − R ) the fol lowing ine quality holds b C ( ρ ) ≥ r − J ( t r , ρ ′ ) + 1 2 ln (1 + ρ ) (1 + ρ ′ ) + o (1) . (63) Using the relation (63 ) in the inequalit y (38) w e pro v e theorem 2. W e ha v e 1 n ln 1 P e ≤ min r ≤ R max ρ ′ ≥ τ r  A (1 − ρ ) 4 − b ( ρ )  + o (1) ≤ ≤ min r ≤ R max ρ ′ ≥ τ r  A (1 − ρ ′ ) e 2( r − R ) 4 − r + J ( t r , ρ ′ ) + 1 2 ln 1 + ρ ′ 1 + ρ  = min r ≤ R max ρ ≥ τ r f ( r , ρ ) , (64) 17 where f ( r , ρ ) = A (1 − ρ ) e 2( r − R ) 4 + R − 2 r + J ( t r , ρ ) + 1 2 ln 1 + ρ 2 e 2( R − r ) + ρ − 1 . With t = t r and (1 − τ r ) e 2( r − R ) = 2 z w e ha v e f ′ ρ = − Ae 2( r − R ) 4 − 1 2(2 e 2( R − r ) + ρ − 1) + 4 t (1 + t ) ρ + p (1 + 2 t ) 2 ρ 2 − 4 t (1 + t ) + 1 2(1 + ρ ) , f ′ ρ   ρ = τ r = − Ae 2( r − R ) 4 − 1 2(2 e 2( R − r ) + τ r − 1) + 1 2(1 − τ r ) = = Az 2 − ( A + 2) z + 1 2(1 − z )(1 − τ r ) , f ′′ ρρ < 0 . Sine f ′′ ρρ < 0 then ρ = τ r is optimal if f ′ ρ   ρ = τ r ≤ 0 . Sine r ≤ R then z ≤ 1 . Therefore f ′ ρ   ρ = τ r ≤ 0 if the follo wing inequalities are fullled: 2 A + 2 + √ A 2 + 4 ≤ z ≤ A + 2 + √ A 2 + 4 2 A . (65) The righ t one of the inequalities (65) is alw a ys satised. The left one of the inequalities (65 ) is equiv alen t to the inequalit y f 2 ( r ) = 2 r + ln(1 − τ r ) ≥ 2 R − 2 R crit ( A ) . (66) The next simple te hnial lemma onerns the funtion f 2 ( r ) in the left-hand side of (66 ). L e m m a 3. The funtion f 2 ( r ) fr om (66 ) monotone de r e ases on 0 ≤ r < R 2 , and monotone inr e ases on r > R 2 , wher e R 2 is dene d in (14 ). Mor e over, the formula holds ln (1 − τ 1 ( A )) = − 2 R crit ( A ) , A > 0 . (67) Sine the funtion E ( R , A ) , R ≥ R 1 ( A ) , is kno wn exatly (see theorem 1), w e onsider only the ase R < R 1 ( A ) . Then t w o ases are p ossible: R ≤ min { R 1 ( A ) , R 2 } and R 2 < R < R 1 ( A ) . C a s e R ≤ min { R 1 ( A ) , R 2 } . F or R ≤ R 2 minim um (o v er r ≤ R ) in the left-hand side of (66 ) is attained when r = R , and then due to (67) the inequalit y (66) redues to the ondition τ R ≤ τ 1 ( A ) , i.e. to R ≤ R 1 ( A ) . Therefore if r ≤ R ≤ min { R 1 ( A ) , R 2 } then the inequalities (66 ) and (65 ) are fullled, and then ρ = τ r is optimal in the righ t-hand side of (64 ). Sine J ( t r , τ r ) = ln(1 + 2 t r ) = − ln(1 − τ 2 r ) / 2 (see (51) and (7 )), then (64 ) tak es the form 1 n ln 1 P e ≤ min r ≤ R f ( r , τ r ) = min r ≤ R C ( v ( r )) − R , R ≤ min { R 1 ( A ) , R 2 } , (68) 18 where C ( v ) = Av 4 − 1 2 ln[ v (2 − v )] , v ( r ) = (1 − τ r ) e 2( r − R ) . (69) Note that for r = R the inequalit y (68 ) redues to the previous b ound ( 10). W e sho w that su h r is optimal in ( 68 ). W e ha v e 4 v (2 − v ) C ′ v = − Av 2 + 2( A + 2) v − 4 , C ′′ v 2 > 0 . Sine 0 ≤ v ≤ 1 , the equation C ′ v = 0 has the unique ro ot v 1 , where v 1 = 4 A + 2 + √ A 2 + 4 = e − 2 R crit ( A ) . (70) The funtion C ( v ) , 0 ≤ v ≤ 1 , monotone dereases on 0 ≤ v < v 1 and monotone inreases on v > v 1 . Note that sine v ( r ) = e f 2 ( r ) − 2 R , then (see lemma 3) the funtion v ( r ) monotone dereases on 0 ≤ r < R 2 and monotone inreases on r > R 2 . If no w R ≤ min { R 1 ( A ) , R 2 } , then v ( r ) ≥ v 1 for r ≤ R . Therefore r = R is optimal in (68 ), and then (68 ) redues to the previous b ound (10 ). C a s e R 2 < R < R 1 ( A ) (i.e. A > A 0 ). Then R 2 < R 3 ( A ) < R 1 ( A ) , where R 3 ( A ) is dened in (14). Consider rst the ase R 2 ≤ R ≤ R 3 ( A ) . It is simple to  he k that then the inequalit y (66) is again satised (see ( 14)). Therefore ρ = τ r is optimal in the righ t-hand side of (64 ), and (64) tak es the form (68). Sine R ≤ R 3 ( A ) , then v ( r ) ≥ v 1 for r ≤ R . Sine R ≥ R 2 then r = R 2 is optimal in (68), and then from (68) the seond of b ounds (16 ) follo ws. It remains to onsider the ase R 2 ≤ R 3 ( A ) ≤ R ≤ R 1 ( A ) . Sine minim um of C ( v ) o v er 0 ≤ v ≤ 1 is attained for v = v 1 (see (70 )), then min 0 ≤ v ≤ 1 C ( v ) = C ( v 1 ) = E sp ( R crit , A ) + R crit , (71) where the form ula w as used E sp ( R crit , A ) + R crit = Av 1 4 − 1 2 ln v 1 − 1 2 ln(2 − v 1 ) . No w in the righ t-hand side of (64) w e set r su h that v ( r ) = v 1 (it is p ossible when R ≥ R 3 ). Then again the inequalit y (66) is fullled and ρ = τ r is optimal in the righ t-hand side of (64 ). F rom (68) and (71) the rst of upp er b ounds ( 15 ) follo ws. The upp er b ound (15 ) an also b e pro v ed applying the straigh t-line b ound to the sphere-pa king b ound and the seond of upp er b ounds (16) at R = R 3 , and the form ula E sp ( R crit , A ) + R crit − R 3 = Aae − 2 R 3 4 − 1 2 ln(2 − ae − 2 R 3 ) − 1 2 ln a , whi h is simple to  he k using the relations ( 12). It ompletes the pro of of theorem 2. N 19 APPENDIX P r o o f o f f o r m u l a (32). Without loss of generalit y w e ma y assume that x i , x j , y ha v e the form x i = ( x 1 , x 2 , 0 , . . . , 0) , x j = ( − x 1 , x 2 , 0 , . . . , 0) , y = (0 , y 2 , y 3 , . . . , y n ) , from whi h w e ha v e d ( x i , x j ) = 4 x 2 1 = 2 An (1 − ρ ij ) , d ( x i , y ) = x 2 1 + ( y 2 − x 2 ) 2 + n X k =3 y 2 k = sn , x 2 1 + x 2 2 = An , n X k =2 y 2 k = r n . Solving those equations w e get x 1 = r An (1 − ρ ij ) 2 , x 2 = r An (1 + ρ ij ) 2 , y 2 = ( A + r − s ) n p 2 An (1 + ρ ij ) , (72) and therefore n X k =3 y 2 k = r n − y 2 2 = r n − ( A + r − s ) 2 n 2 A (1 + ρ ij ) = r 1 n , from whi h the form ula ( 32 ) follo ws. N Optimalit y of s ( ρ ) , r ( ρ ) from the form ulas (37) also follo ws from (72). P r o o f o f f o r m u l a ( 53 ). F or the funtion f ( ρ ) = J ( t R , ρ ) − Aρ/ 4 from (51 ) w e ha v e f ′ = 4 t R (1 + t R ) ρ + p (1 + 2 t R ) 2 ρ 2 − 4 t R (1 + t R ) − A 4 , f ′′ ( t, ρ ) < 0 . Then for ρ ≥ τ R w e ha v e f ′ ≤ f ′    ρ = τ R = 4 t R (1 + t R ) τ R − A 4 = τ R 1 − τ 2 R − A 4 ≤ 0 , if τ R ≤ τ 1 ( A ) , whi h pro v es the form ula (53 ). N P r o o f o f l e m m a 1. Let { x 1 , . . . , x M } ⊂ S n − 1 ( √ An ) b e a o de su h that max i 6 = j ( x i , x j ) ≤ 0 , i.e. min i 6 = j k x i − x j k 2 ≥ 2 A . Then, learly , M ≤ 2 n . In lemma 1 for all i w e ha v e ( x i , y ) = ( A + r − s ) n/ 2 . Consider M v etors { x ′ i = x i − a y } , where a = ( A + r − s ) / (2 r ) . Then due to the ondition ( 48) w e ha v e max i 6 = j  x ′ i , x ′ j  ≤  4 Ar ρ − ( A + r − s ) 2  n/ (4 r ) ≤ 0 , 20 and therefore M ≤ 2 n . N P r o o f o f l e m m a 2. T o pro v e lemma w e redue it to the ase ρ ≈ 0 , and then use lemma 4 (see b elo w). W e set some in teger m su h that 1 < m < M , and in tro due the v etor z = a m X k =1 x k , a = ρ 1 + ( m − 1) ρ . After simple alulations w e get ρ − δ − 1 m ≤ k z k 2 ≤ ρ + δ , ρ − δ 1 + (1 − ρ ) / ( mρ ) ≤ ( x i , z ) ≤ ρ + δ 1 + (1 − ρ ) / ( mρ ) , i = m + 1 , . . . , M . (73) Consider the normalized v etors u i = x i − z k x i − z k , i = m + 1 , . . . , M . Using the form ulas (73), for an y i, j ≥ m + 1 , i 6 = j , w e get ( u i , u j ) ≤ 2 (1 − ρ )  δ + 1 m  = o (1) , n → ∞ , (74) if w e set m → ∞ as n → ∞ . T o upp erb ound the maximal p ossible n um b er M − m of v etors { u i } satisfying the ondition (74 ), w e use a mo diation of [16, Theorem 2℄. L e m m a 4. L et C = { x 1 , . . . , x M } ⊂ S n − 1 (1) b e a  o de with ( x i , x j ) ≤ µ, i 6 = j . Then for n ≥ 1 the upp er b ound holds M ≤ 2 n 3 / 2 (1 − µ ) − n/ 2 , 0 ≤ µ < 1 . (75) P r o o f. Denote µ = cos(2 ϕ ) , and let M ( ϕ ) b e the maximal ardinalit y of su h a o de. F or M ( ϕ ) the upp er b ound holds [ 16 , Theorem 2℄ M ( ϕ ) ≤ ( n − 1) √ π Γ  n − 1 2  sin β ta n β 2Γ  n 2   sin n − 1 β − f ( β , n − 2) cos β  , 0 < ϕ < π 4 , (76) where β = arcsin( √ 2 sin ϕ ) and f ( β , n − 2) = ( n − 1) β Z 0 sin n − 2 z dz . 21 In tegrating b y parts, for the funtion f ( β , n − 2) w e ha v e f ( β , n − 2) = sin n − 1 β cos β − sin n +1 β ( n + 1) cos 3 β − 3 ( n + 1) β Z 0 sin n +2 z cos 4 z dz ≥ ≥ sin n − 1 β cos β − sin n +1 β ( n + 1) cos 3 β − 3 tan 4 β ( n + 1) f ( β , n − 2) , and therefore 1 .  1 + 3 tan 4 β n 2 − 1  ≤ f ( β , n − 2) .  sin n − 1 β cos β  1 − tan 2 β n + 1  ≤ 1 , (77) if tan 2 β < n + 1 , i.e. if 2 sin 2 ϕ < ( n + 1) / ( n + 2) . F rom ( 76) and (77 ) w e get M ( ϕ ) ≤ √ π Γ  n − 1 2  ( n 2 − 1) cos β 2Γ  n 2  sin n − 1 β < n p π n (1 − 2 sin 2 ϕ ) √ 2  √ 2 sin ϕ  n − 1 , (78) sine Γ  z − 1 2  ( z 2 − 1 )  Γ  z 2  < √ 2 z 3 / 2 e 1 /z , z ≥ 0 . F rom (78 ) the inequalit y (75) follo ws pro vided 2 sin 2 ϕ < ( n + 1) / ( n + 2) , i.e. if µ > 1 / ( n + 2) . Sine the funtion M ( ϕ ) is on tin uous on the left for ϕ ∈ (0 , π ] , the upp er b ound (78 ) remains v alid for µ = 1 / ( n + 2) as w ell. F or µ = 1 / ( n + 2) , n ≥ 1 , the righ t-hand side of (78 ) do es not exeed n p π e/ 2 , whi h in turn do es not exeed the righ t-hand side of (75 ) for an y µ ≥ 0 , n ≥ 2 . Sine M ( ϕ ) is a dereasing funtion, it pro v es the inequalit y (75 ) for an y µ ≥ 0 , n ≥ 2 . Clearly , (75 ) remains v alid for n = 1 as w ell. N No w from (74 ) and (75) w e get lemma 2. N The author thanks L.A.Bassalygo, G.A.Kabat y ansky and V.V.Prelo v for useful disussions and onstrutiv e ritial remarks. 22 REFERENCES 1. Shannon C. E. Probabilit y of Error for Optimal Co des in Gaussian Channel // Bell System T e hn. J. 1959. V. 38.  3. P . 611656. 2. Shannon C. E., Gal lager R. G.. Berlekamp E. R. 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