Some Unitary Space Time Codes From Sphere Packing Theory With Optimal Diversity Product of Code Size

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER Some Unitary Sace Time Codes From Shere Packing Theory With Otimal Diversity Product of Code Size Haiquan Wang, Genyuan Wang, and Xiang-Gen Xia, Senior Member, IEEE Abstract In this corresondence, we roose some new designs of unitary sace time codes of sizes 6, 3, 48, 64 with best-known diversity roducts (or roduct distances) by artially using shere acking theory. In articular, we resent an otimal unitary sace time code of size 6 in the sense that it reaches the maximal ossible diversity roduct for unitary sace time codes of size 6. The construction and the otimality of the code of size 6 rovide the recise value of the maximal diversity roduct of a unitary sace time code of size 6. Index Terms Differential sace time modulation, otimal diversity roduct, acking theory, unitary sace time codes. I. INTRODUCTION Unitary sace time codes have been recently roosed in [6], [5] for differential sace time modulation schemes and in [] [4] for ossibly other sace time modulation schemes. Unitary sace time codes in differential encoding are useful not only when the channel information is not known at the receiver and noncoherent decoding is used but also when the channel information is known at the receiver and coherent decoding as a recursive trellis coding is used jointly with an error correction coding as a turbo tye coding [9] where a suer erformance is achieved. There have been several unitary sace time code constructions in the literature: for examle, grou and otimal grou constructions [6], [7], [5], [9]; orthogonal designs [8]; arametric codes []; Cayley transforms []; Lie grous [3], [6]; and Hamiltonian constellations or sherical codes using acking theory [9], [6]. It is known that the erformance of a sace time code deends on its diversity roduct and having a good diversity roduct has become an imortant criterion in the design of a sace time code. In [], some uer bounds on the diversity roducts of (unitary) sace time codes for a given size are resented. It is easy to reach the diversity roduct uer bound for matrices of sizes below 4 and unitary matrices of sizes 4 and 5 reaching the uer bound are also resented in [] using the arametric forms of unitary matrices. In fact, unitary matrices of sizes below 6 reaching the uer bound can be also constructed by using the Hamiltonian constellations from the acking theory, i.e., the otimal shere acking oints. However, in [] it is shown that the uer bound is not reachable when the unitary code size is above 5 and a tight uer bound on the diversity roducts remains oen. The otimal or best-known shere acking oints of sizes above 5 do not rovide otimal unitary sace time codes with otimal diversity roducts. Manuscrit received March 6, 3; revised June 5, 4. This work was suorted in art by the Air Force Office of Scientific Research under Grant F and the National Science Foundation under Grants CCR- 974 and CCR-358. The material in this corresondence was resented in art at the IEEE International Symosium on Information Theory, Yokohama, Jaan, June /July. The authors are with the Deartment of Electrical and Comuter Engineering, University of Delaware, Newark, DE 976 USA ( hwang@ ece.udel.edu; gwang@ece.udel.edu; xxia@ece.udel.edu). Communicated by G. Caire, Associate Editor for Communications. Digital Object Identifier.9/TIT In this corresondence, we roose some unitary sace time codes by artially using the otimal shere acking oints [], []. We obtain a determinant relationshi for difference matrices between Hamiltonian and general unitary constellations. We resent some best-known designs for size L =6; 3; 48; 64, and also show that the code with size 6 reaches the otimal diversity roduct. This corresondence is organized as follows. In Section II, we resent new best-known diversity roduct designs for size L = 6; 3; 48; 64. In Section III, we show the otimality of the new code of size 6 resented in Section II. Since the roof is highly technical, we leave the most technical arts of the roofs in [3], which is downloadable via our website. II. SOME UNITARY CODES WITH BEST KNOWN DIVERSITY PRODUCTS In this section, we resent some new unitary codes for sizes L = 6; 3; 48; 64 with best-known diversity roducts. A. Diversity Product Let G = fv ;V ;...;V Lg be a unitary sace time code of size L with Vl H V l = I where H stands for the transose and comlex conjugate. Define and d L max G (G) min j det(v l V l )j () V ;V G;l6=l (G) = max G min j det(v l V l )j: () V ;V G;l6=l Following the convention in the literature, the diversity roduct for a code G is defined as follows: (G) (G) (3) and the otimal diversity roduct for L-oint constellation is defined as (L) max (G) = d L: (4) G We are interested in designing a code G with large or otimal diversity roduct. B. Unitary Matrices The content resented here can be found in many literatures, for examle, [], [6]. For the notational convenience for our later study, we briefly introduce some concets on unitary matrices below. Let U () be the set of all unitary matrices, i.e. U () fa j A is a matrix with A H A = Ig: Between U () and the unit ball S 3 4, there exists a close relationshi as follows. For any matrix A with A H A = I, wehavej det(a)j = and thus there is a unique angle [; ) such that det(a) =e j. For any fixed angle [; ), let Thus, we have SU(;) fa U () j det(a) =e j g: (5) U () = [;) SU(;): /4$. 4 IEEE

2 336 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER 4 We are articularly interested in the case of =and denote the set SU(; ) by SU() for short, i.e., SU() = SU(; ). We now investigate the structure of SU(). From some theory of unitary matrices (for examle, see []), SU() can be isometrically embedded onto the four-dimensional (4-D) Euclidean real unit shere. Let S 3 be the unit shere of 4-D real Euclidean sace 4, i.e. S 3 = fx 4 jkxk =g where kkdenotes the conventional l norm. Because det(a) =and the unitariness for any element A in SU(), it is not hard to see that there are two comlex numbers a = a + ja and b = b + jb such that a b A = b 3 a 3 (6) where 3 denotes the conjugate, and a ;a ;b ;b are real numbers governed by the condition a + a + b + b =, i.e., jaj + jbj =, []. From this exression, the following embedding from SU() onto S 3 can be obtained, also see for examle [6]. Let i be the maing i : A 7! i(a) from SU() into S 3 defined by i(a) (a ;a ;b ;b )=(Re(a); Im(a); Re(b); Im(b)) (7) where a ;a ;b ;b are the real numbers defined in (6) and Re and Im stand for the real and imaginary arts of a comlex number, resectively. Clearly, the maing i is one-to-one and onto. Furthermore, the following relationshi holds: det(a B) =ki(a) i(b)k : (8) This equation also imlies that all determinants of difference matrices of two distinct unitary matrices in SU() are ositive. From (8), one can see that the roblem to find an otimal sace time code in SU(), i.e., it is restricted to have determinant, becomes to find otimal acking oints on the shere S 3, which is called Hamiltonian constellations in [9]. Thus, as indicated in [9], if we denote D L as the maximal minimum-distance of L-oint acking on S 3, then d L D L i.e., the square of the maximal minimum-distance of L-oint acking on S 3 is a lower bound for d L. However, as we shall see later, the above Hamiltonian constellation may not be enough to have good codes and we need to consider the entire unitary matrix sace U ().Todo so, we need a determinant formula. C. A Useful Determinant Formula Let us consider a relationshi between SU() and U () or equivalently between SU() and SU(;) for any [; ). For a fixed, we define a maing J from SU(;) to SU() as follows: J (A) e j= A; for A SU(;): (9) Since det(j (A)) = e j det(a) =e j e j =, this maing is well defined. Furthermore, it is not hard to see that it is one-to-one and onto. With this notation, one can see that any unitary matrix A can be reresented by A = e j= J (A); for some [; ): An imortant roerty from this maing is that it also rovides a determinant formula for a difference matrix of two matrices selected from different sets SU(; ) and SU(; ), which is stated in the following roosition. Proosition : For any A SU() and A SU(;),wehave j det(a A )j = j det(a J (A)) 4sin (=4)j: Proof: Assume A = a b b 3 a 3 and J (A) = where ja j + jb j =and jaj + jbj =. Then a b b 3 a 3 a b a b det(a J (A)) = det( b 3 a 3 b 3 a 3 ) = (a a 3 + b b 3 + a 3 a + bb): 3 By the definition of J in (5), we have A = e j= J (A). Therefore, j det(a A)j a b = det b 3 a 3 e j= a e j= b e j= b 3 e j= a 3 = j +e j e j= (a a 3 + b b 3 + a 3 a + bb)j 3 = j +e j e j= ( det(a J (A)))j = je j= + e j= ( det(a J (A)))j which is the same as the one in the roosition. QED From this roosition, we immediately have the following corollary. Corollary : For any A SU(; ) and A SU(; ),we have j det(a A )j =j det(j (A )J (A ))4 sin (( )=4)j: From the above roosition and corollary, one can see that the determinant absolute value of the difference matrix of two unitary matrices deends on the distance between their embeddings and their angle difference. This motivates us to design a unitary sace time code using two stes: one is to select good acking oints on the shere S 3 and the other is to select good angles. D. Some New Codes With Best Known Diversity Products With the hel of the above determinant formulas, we can construct some unitary codes with best-known diversity roducts. ) Size L =6: Let d = 5=+. Select a four-oint acking on S 3 as follows: a =(a; b; b; b); a =(a; b; b; b) a 3 =(a; b; b; b); a 4 =(a; b; b; b) where a = 3d=8 and b = ( a )=3. By maing these oints back to SU(), we have the following four unitary matrices: and A = A = A 3 = A 4 = a bj b bj b bj a + bj a + bj b + bj b + bj a bj a bj b + bj b + bj a + bj a + bj b bj b bj a bj For other two unitary matrices, we use angle. Let = arccos(d= a) and = A 5 = e j = I SU(; ); A 6 = e j : = I SU(; ):

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER It is easy to check that the diversity roduct of the code fa ; A ;...;A 6g is In next section, we shall rove that 5= + :74: (6) = 5= + i.e., this code reaches the otimal diversity roduct of any unitary sace time codes of size L =6. ) Sizes L = 3; 48; 64: To construct 3; 48; or 64 unitary matrices with large diversity roducts, at first, we first construct four diamonds in S 3 as follows. Let t is a arameter, and a = 3 8 t ; r = 6 b = t; r = a 3 3 t; = 3 : The four-oint coordinates of the first diamond are a =(a; r; ; ); a =(a; b; r ; ) a 3 =(a; b; r cos();r sin()); a 4 =(a; b; r cos();r sin()): The ones of the second diamond are a 5 =(a; r; ; ); a 6 =(a; b; r ; ) a 7 =(a; b; r cos(); r sin()) a 8 =(a; b; r cos(); r sin()): The ones of the third diamond are a 9 =(a; r; ; ); a =(a; b; r ; ) a =(a; b; r cos();r sin()); a =(a; b; r cos();r sin()): The ones of the fourth diamond are a 3 =(a; r; ; ); a 4 =(a; b; r ; ) a 5 =(a; b; r cos(); r sin()) a 6 =(a; b; r cos(); r sin()): Maing these oints back to SU() using the ma i given in (7), we obtain 6 matrices, denoted by Q j, i.e., Q j i (a j ) for j = ; ;...; 6. These matrices can be used to generate best-known diversity roduct unitary codes with L =3; 48; and 64 as follows. For L =3, let t =, = arccos(3=4) and define U i = Q i ; i =; ; 3; 4 U i = Q 8+i; i =5; 6; 7; 8 U i = e j(=4+=) Q i8; i =9; ; ; U i = e j(=4+=) Q i; i =3; 4; 5; 6 U i = e j(=) Q i; i =7; 8; 9; U i = e j(=) Q i; i =; ; 3; 4 U i = e j(3=4+=) Q i; i =5; 6; 7; 8 U i = e j(3=4+=) Q i; i =9; 3; 3; 3 and ut G 3 = fu ;...;U 3g, then the minimum determinant (G 7 3) =, and the diversity roduct (G 3) is 7 which is best known for size L =3. For L =48, let t = and = :4536 V i = Q i; i =; ; 3; 4; V i = Q 8+i ; i =5; 6; 7; 8 V i = e j(=6) Q i4; i =9; ; ; V i = e j(=6) Q i4; i =3; 4; 5; 6 V i = e j(=3) Q i6; i =7; 8; 9; V i = e j(=6) Q i8; i =; ; 3; 4 V i = e j(=) Q i; i =5; 6; 7; 8 V i = e j(=) Q i; i =9; 3; 3; 3 V i = e j(=3) Q i3; i =33; 34; 35; 36 V i = e j(=3) Q i4; i =37; 38; 39; 4 V i = e j(5=6) Q i36; i =4; 4; 43; 44 V i = e j(5=6) Q i36; i =45; 46; 47; 48 and define G 48 = fv ;...;V 48g, then the minimum determinant (G 48 )= 3, and the diversity roduct (G 48 ) is 3 = :478 which is best known for size L =48. For L =64, let t = :388, and define W i = Q i ; i =; ; 3; 4 W i = Q 8+i; i =5; 6; 7; 8 W i = e j(=8) Q i4; i =9; ; ; W i = e j(=8) Q i4; i =3; 4; 5; 6 W i = e j(=4) Q i6; i =7; 8; 9; W i = e j(=4) Q i8; i =; ; 3; 4 W i = e j(3=8) Q i; i =5; 6; 7; 8 W i = e j(3=8) Q i; i =9; 3; 3; 3 W i = e j(=) Q i3; i =33; 34; 35; 36 W i = e j(=) Q i4; i =37; 38; 39; 4 W i = e j(5=8) Q i36; i =4; 4; 43; 44 W i = e j(5=8) Q i36; i =45; 46; 47; 48 W i = e j(3=4) Q i48; i =49; 5; 5; 5 W i = e j(3=4) Q i4; i =53; 54; 55; 56; W i = e j(7=8) Q i5; i =57; 58; 59; 6 W i = e j(7=8) Q i5; i =6; 6; 63; 64 and define G 64 = fw ;...;W 64 g, then the minimum determinant (G 64 )=:546, and the diversity roduct (G 64 ) is :546 = :3676 which is best known for size L =64.

4 3364 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER 4 TABLE I DIVERSITY PRODUCT AND SUM COMPARISONS Fig.. SER comarison. Table I summarizes the above results and comares with some existing codes, where diversity sum means the minimum Euclidean distance between codeword matrices []. From Table I, one can see that the otimal diversity sum :7746 of the unitary code of size 6 resented in [] is slightly better than the one :74 of the unitary code of size 6 with otimal diversity roduct resented in this aer. Fig. shows the symbol error rates (SER) of these two codes of size 6 over a quasi-static fading channel and one can see that the one with the otimal diversity roduct erforms slightly better than the one with the otimal diversity sum at high signal-to-noise ratio (SNR), which also confirms the argument between diversity roduct and diversity sum in []. III. OPTIMALITY OF UNITARY SPACE TIME CODES OF SIZE L =6 The main goal of this section is to rove the otimality of the code of size 6 resented in Section II-D-). Theorem : The maximal diversity roduct of a unitary sace time code of size 6 is 5= +, i.e., (6) = 5= + : This theorem imlies that the code resented in Section II-D-) has already reached the maximal diversity roduct. To rove this theorem, we need some rearations. First, we introduce the concet of dual. For any unitary matrix A = e j= J (A), its dual is defined as e j()= (J (A)) and denoted by A.If ~ =, then A ~ = e j= (J (A)) = A, i.e., the dual of A SU() is itself. With the definition of a dual matrix we have the following corollary. Lemma : For any unitary matrices A and A with their duals A ~ and A ~, resectively, we have i) j det(a B)j = j det( A ~ B)j, for any unitary matrix B SU(), ii) j det(a A )j = j det( A ~ A ~ )j.

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER This lemma is a direct result of Proosition, we omit its roof. In what follows, for the notational convenience, we use ^A to denote J (A) for a matrix A SU(;) by droing the subscrit without causing any confusion. Since there exists an embedding i from SU() onto S 3, we do not distinguish a matrix in SU() and a vector on S 3 and use the same notation A to exress a matrix in SU() and a oint on S 3.IfA is treated as a oint on S 3, it means its embedding, i.e., i(a) in (7). Lemma : Let A i SU(; i ), i = ; ;...;L, and fa ;A ;...;A Lg be an otimal unitary sace time code of size L with the maximal diversity roduct d L. Then, j det(a i A j )j = det( ^A i ^A j ) 4sin (( i j )=4) d L ; if j i j j j det(a i A j )j = 4 sin (( i j )=4) det( ^A i ^A j ) d L ; if j i j j where ^A l = J (A l ) is the rojection of A l from SU(; l ) to SU() as defined in Section II-B. Its roof is in [3]. Lemma basically rovides an exression of the absolute value of a difference matrix determinant from the one of their rojections to SU() and their angles for an otimal constellation. Lemma 3: Let fa ;...;A L g be an unitary sace time code with the otimal diversity roduct d L > and A j SU(; j ), j = ; ;...;L;with 6.If= L <, then i+ i < for i =; ;...;L, i.e., the difference of two adjacent angles is less than. Its roof is in [3]. Lemma 4: Let fp ;...;P L g be L oints on the shere S 3. Assume that kp i P j k d for a constant d and i<j L. Let P be any a oint on this shere. Then if L =4, there exist s and t, s; t L, such that kp P s k 3d=8 and kp P t k + 3d=8 if L =3, there exist s and t, s; t L, such that kp P s k d=3 and kp P t k + d=3 if L =, there exist s and t, s; t L, such that Its roof is in [3]. kp P s k d=4 and kp P tk + d=4: Lemma 5: For any L oints fp ;...;P Lg on the unit shere S n in the n +-dimensional real Euclidean sace n+,wehave i<jl kp i P j k L : Its roof can be found in, for examle, []. Lemma 6: Let A i = e j = ^Ai SU(; i ), i =; ; 3, be three unitary matrices. Assume 3.If 3, and for i 6= j, j det(a i A j )jd 6 5=, then, 3 5=6: Its roof is in [3]. Lemma 7: Let < d :5 and < a;b d=. If arccos(d= + a) + arccos(d= + b) =, then sin(arccos(a + d=)=) b cos(arccos(d=+b) + arccos(d= +a)) d= d; cos(arccos(a)=) () where arccos(x). Its roof is in [3]. Proosition : Let fa ;A ;...;A 6 g be an otimal constellation with A j = e j = ^Aj of the maximal diversity roduct d 6. Assume that = 5 6 and 6 5, 6 4. Then, d 6 5= +. Its roof is in [3]. Proosition 3: Let fa ;A ;...;A 6 g be an otimal constellation with A j = e j = ^Aj. Assume that = 5 6 and 6 4, 6 3. Then, d 6 < 5= +. Its roof is in [3]. Proosition 4: Let fa ;A ;...;A 6 g be an otimal constellation with A j = e j = ^Aj. Assume that = and 6 4, 6 3. Then, d 6 < 5= +. Its roof is in [3]. Proosition 5: Let fa ;A ;...;A 6g be an otimal constellation with A j = e j = ^Aj. Assume that = and 6 3, 6. Then, d 6 < 5= +. Its roof is in [3]. Now we begin to rove Theorem. Proof of Theorem : Assume signal constellation G = fa ;A ;...;A 6g is an otimal constellation with the maximal diversity roduct d 6 and A i SU(; i ), i =; ;...; 6. By the construction in Section II-D ), we have d 6 5= +. We next need to show that d 6 5=+. To do so, let us consider the different cases of the number of the zero angles of A i : ]fi j i =g. Without loss of generality, we can assume that 6. In this roof and the roofs in [3], we always use arccos(x). i) = 6. =6means that all A i SU(), i.e., all six matrices A i are on the shere S 3. In other words, there exist 6-oint acking such that the minimal distance is greater than, which contradicts with the acking result on S 3 (according to the result [], the acking angle on S 3 is =, that is, the maximal minimum distance is ). ii) = 5. Assume = = 5 = and 6 >. Thus, A i = ^A i;i = ; ;...; 5. By Lemma 3, we have 6 5, i.e., 6. By Lemma det( ^A i ^A 6 ) d sin ( 6 =4) > ; i =; ;...; 5: For i 6= j 5, from the condition, det( ^A i ^A j ) >. Therefore, there exists six oints f ^A ;...; ^A 6 g on the shere S 3 such that the minimum distance is greater than, which contradicts with the acking result as in (i). iii) =4.

6 3366 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER 4 Assume = = 4 =and < 5 6 <. By Lemma 3, we have 5 and 6 5. If 6, then, as shown as ii), f ^A ;...; ^A 4 ; ^A 5 ; ^A 6 g consists of a 6-oint acking on S 3 with minimum distance greater than, which results in a contradiction. Therefore, we can assume that 5 6 and 6 5. We next investigate the acking osition of f ^A ;...; ^A 4 ; ^A 5 ; ^A 6 g on S 3. Denote ^A i =(a i;b i;c i;e i); i=;...; 6. By a unitary transformation, we can assume ^A 5 = I, i.e., a 5 =;b 5 = c 5 = e 5 =. We then convert this roblem to a acking roblem on the 3-D unit shere S as follows. If a i 6=;, define where r i = b i = r i (b i ;c i ;e i ); i =; ; 3; 4; 6 () a i. Then b i S and clearly det( ^A i ^A j ) = ( a i a j )+r i r j (kb i b j k ) () kb i b j k = ( aiaj ) det( ^A i ^A j ) r ir j : (3) Two remarks about this conversion are as follows. The maing S 3 3 (a; b; c; e)! b = (b=r; c=r; e=r) S is not one-to-one. It is because, for different two oints (a; b; c; e) and (a; b; c; e), the images are the same. However, when we restrict a or a, the maing becomes one-to-one and onto. Another remark is that an image oint b does not deend on a, when we restrict a to a or a. To exlain this, we use the olar coordination. For any oint (a; b; c; e) S 3, there exist three angles ; ; 3, such that a =sin( ) and b = cos( ) sin( ), c = cos( ) cos( ) sin( 3), e = cos( ) cos( ) cos( 3). Hence b =(b=r; c=r; e=r) = (sin( ), cos( ) sin( 3 ), cos( ) cos( 3 )), which is indeendent of, i.e., a. Therefore, when we restrict a to a or a, the distance kb i b j k is indeendent of a i and a j. For i 4 and i =6, because 5 and 6 5, by Lemma, we have <d 6 jdet(a 5 A i )j = det( ^A 5 ^A i ) 4 sin ( 5 =4) = a i 4 sin ( 5 =4): Therefore, a i < for i =; ; 3; 4; 6. Since = = 3 = 4 =, without loss of generality, we may assume that a a a 3 a 4 <. Ifa =, then b = c = e =and it is not hard to see that det( ^A 4 ^A )=+a 4.It imlies d 6 +a 4, i.e., a 4 >, which contradicts with the result a 4 < we derived before. Therefore, a >. For i 6= j 4, from () and the fact that i = j =,we have d 6 jdet(a i A j )j = det( ^A i ^A j ) 4 sin (( i j)=4) = a ia j + r ir j(kb i b jk ): Because fb ;...;b 4g are on S, by the acking theory on S, there is at least one air fb i ;b j g such that kb i b j k 8=3. Hence, a 4 3d 6=8: (4) Since a 5 =; a 6 <, and 6 5, from Lemma we have d 6 jdet(a 6 A 5)j = det( ^A 6 ^A 5) 4 sin (( 6 5)=4) = a 6 4 sin (( 6 5)=4) () 4 sin (( 6 5 )=4): Therefore, Using the fact that cos(( 6 5 )=) d 6 = : (5) d 6 jdet(a 5 A 4 )j = det( ^A 5 ^A 4 ) 4 sin ( 5 =4) and noting that det( ^A 5 ^A 4)= a 4,wehave d 6 a 44 sin ( 5=4) + 3d 6=84 sin ( 5=4) = cos( 5=)+ 3d 6=8 (6) where the second inequality is from (4). Inequality (6) imlies cos( 5 =) d 6 = 3d 6 =8: (7) We now relace A 5 ;A 6 by their duals ~ A 5 ; ~ A 6. From the definition, we have ~A 6 = e j( )= ( ^A 6 ); ~ A5 = e j( )= ( ^A 5 ): Furthermore, fa ;...;A 4 ; ~ A 6 ; ~ A 5 g is also an otimal signal constellation by Lemma. We make a normalization by multilying ^A H 6 from left to the constellation to get a new constellation G f ^A H 6 A ; ^A H 6 A ; ^A H 6 A 3 ; ^A H 6 A 4 ; ^A H 6 ~ A 6 ; ^A H 6 ~ A 5 g = f ^A H 6 A ; ^A H 6 A ; ^A H 6 A 3 ; ^A H 6 A 4 ;e j( )= I; ^A H 6 A ~ 5 g: Since ^A H 6 is a unitary matrix, G is also an otimal constellation. Furthermore, G and fa ;...;A 6 g have the same angle relationshis. Therefore, inequality (7) corresonding to this new constellation G also holds cos(( 6 )=) d 6 = 3d 6 =8: (8) From (7) and (8), we have Because a i a j > and r i r j >, wehavekb i b j k >. Furthermore, it is easy to see that for a fixed a i, the right hand side of the above inequality is increasing for a j. Therefore, d 6 a ia j + r ir j(kb i b jk ) a4 + r4(kb i b j k ) Hence, 5 arccos(d 6= 3d 6=8) 6 arccos(d 6= 3d 6=8): arccos(d 6 = 3d 6 =8): which imlies that From (5), we know that 6 5 arccos(d 6 = ). Therefore, a 4 d 6 =kb i b j k : 4 arccos(d 6 = 3d 6 =8) arccos(d 6 = ):

7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER Hence, Thus, if we rearrange ~ G into d 6 4(d 6= 3d 6=8) +4 which imlies the desired result d 6 5= +. iv) 3. Assume = 3 6. Using the same argument as in the beginning of Case (iii) when =4and Lemma 3, we can also assume that 6 and, 6 5. We divide the roof into several cases according to the relationshis among the angles j. Case I 6 and 5. We divide this case into four subcases. Case I. 6, 5 and 6 4. This subcase is Proosition. Case I. 6, 5 and 6 4, 6 3. This subcase is Proosition 3. Case I.3 6, 5 and 6 3, 6. By taking the rotation of angle 5 to G, we obtain a new constellation G = fa ;A ;...;A 6g where A j = e j = A j.forj =; ; 3; 4 A j = e j = e j = ^Aj = e (( ))= ( ^A 5 ): For j =5,wehaveA 5 = e j = A 5 = ^A 5.Forj =6,wehave A 6 = e j = e j = ^A6 = e ( )= ^A6 : Therefore, the relationshi between G and G is and f ^A ; ^A ; ^A 3; ^A 4; ^A 5; ^A 6g =f ^A ; ^A ; ^A 3; ^A 4; ^A 5; ^A 6g; f ; ; 3; 4; 5; 6g = f 5 ; ( 5 ); ( 5 3 ); ( 5 4 ); ; 6 5 g: Clearly, the diversity roduct of G is still d 6. We now consider the dual of G, denoted by G ~, which has the same diversity roduct as G by Corollary : G ~ = f A ~ ; A ~ ;...; A ~ 6g, where ~A j = e = ^~ A, j 6. By the definition of a dual, the relationshi j between G ~ and G or G is f ^~ A ; ^~ A ; ^~ A 3 ; ^~ A 4 ; ^~ A 5 ; ^~ A 6 g = f ^A ; ^A ; ^A 3; ^A 4; ^A 5; ^A 6g = f ^A ; ^A ; ^A 3; ^A 4; ^A 5; ^A 6g and the corresonding angles are f ; ;3 ;4 ;5 ;6 g = f ; ; 3; 4; ; 6g = f 5; 5 ; 5 3; 5 4; ; ( 6 5)g: It is easy to see that 6 and j for j =; ; 3; 4; 5. Furthermore, 6 ; 6 ; 6 j ; j =3; 4; 5: G = f ~ A 5; ~ A 4; ~ A 3; ~ A ; ~ A ; ~ A 6g then the conditions on G are exactly the same as the ones in Case I.. By Proosition 3, we have roved this theorem in this subcase. Case I.4 6, 5 and 6 Make a rotation angle to the constellation G as done in Case I.3 and we find that the new constellation has the same conditions as in Case I.. Therefore, by Proosition, we have roved this theorem in this subcase. Case II 6 ; 5 and 4. We divide this roof into four subcases. Case II. 6 ; 5, 4 and 6 4. In this case, we make a rotation to the constellation as follows. Let A j = e j = A j for j = ; ;...; 6. Then fa ;A ;...;A 6g is also an otimal constellation. Since, for j = ; ; 3; 4; 5, A j = e j = A j = e j(( ))= ( ^A j ), we obtain ^A j = ^A j and the angle j of A j is ( 6 j). Forj = 6, 6 =, i.e., A 6 belongs to SU() and A 6 = ^A 6. Furthermore, we have that 6; ; ; 3; 4 are all less than or equal to, and 5 is greater than or equal to. Therefore, fa ;A ;...;A 6g satisfies the conditions in Case I. Thus we have roved this theorem in this subcase. Case II. 6 ; 5, 4 and 6 4, 6 3. It is roved in Proosition 4. Case II.3 6 ; 5, 4 and 6 3, 6. It is roved in Proosition 5. Case II.4 6; 5, 4 and 6. Let A j = e j = A j for j = ; 3; 4; 5; 6, and A 6 = e (j )= A. Note that ^A = A since =. Then, fa ;A ;A 3;A 4;A 5;A 6g satisfies the conditions of Case I. Case III 6 ; 5 ; 4 and 3. Under this assumtion, we consider the dual constellation: = is fixed, and ; 3 are changed to ; 3, which belong to [; ], and 4 ; 5 : 6 are transferred to 4 ; 5 ; 6, which belong to [;]. Therefore, through this duality, we change this subcase into Case II. Case IV 6 ; 5 ; 4 ; 3 and. Also we consider its dual constellation and find that this case can be converted to Case I. By summarizing all the above cases, this theorem is roved. q:e:d: IV. CONCLUSION In this corresondence, we have artially used shere acking theory to construct unitary sace time codes. Although the otimal ones of sizes L below 6 can be constructed from the shere ackings on S 3, i.e., Hamiltonian constellations [9], [6] that reach the uer bound L=(L ) of the maximal diversity roducts derived in []. This uer bound can not be reached when the sizes are above 5 as shown in []. The critical boundary on the sizes is size L =6. In this corresondence, we have constructed unitary sace time code of size 6 that has been shown in this corresondence to have the otimal diversity roduct. The otimal diversity roduct d 6 = 5= + :74 < :7746 L=(L )

8 3368 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER 4 when L = 6. Some constructions of unitary sace time codes of sizes 3; 48; 64 of non-hamiltonian constellations with best-known diversity roducts have been also resented by artially using shere acking theory. To obtain these results, we have resented a determinant identity between the difference matrices of two matrices in a Hamiltonian constellation and two matrices in non-hamiltonian constellations. Since some of the roofs of the lemmas and roositions in Section III are tedious and long, the longer version of this corresondence including all the roofs is downloadable through the website [3]. [] J. H. Conway and N. J. A. Sloane, Shere Packings, Lattices and Grous, 3rd ed. New York: Sringer-Verlag, 999. [3] H. Wang, G. Wang, and X.-G. Xia. Some unitary sace-time codes from shere acking theory with otimal diversity roduct of code size. [Online]. Available: htt:// REFERENCES [] T. L. Marzetta and B. M. Hochwald, Caacity of a mobile multile-antenna communication link in Rayleigh flat fading, IEEE Trans. Inform. Theory, vol. 45, , Jan [] B. M. Hochwald and T. L. Marzetta, Unitary sace-time modulation for multile-antenna communication in Rayleigh flat fading, IEEE Trans. Inform. Theory, vol. 46, , Mar.. [3] B. M. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens, and R. Urbanke, Systematic design of unitary sace-time constellations, IEEE Trans. Inform. Theory, vol. 46, , Set.. [4] D. Agrawal, T. J. Richardson, and R. L. Urbanke, Multile-antenna signal constellations for fading channels, IEEE Trans. Inform. Theory, vol. 47, , Set.. [5] B. M. Hochwald and W. Sweldens, Differential unitary sace-time modulation, IEEE Trans. Commun., vol. 48,. 4 5, Dec.. [6] B. L. Hughes, Differential sace-time modulation, IEEE Trans. Inform. Theory, vol. 46, , Nov.. [7], Otimal sace-time constellations from grous, IEEE Trans. Inform. Theory, vol. 49,. 4 4, Feb. 3. [8] V. Tarokh and H. Jafarkhani, A differential detection scheme for transmit diversity, IEEE J. Select. Areas Commun., vol. 8, , July. [9] A. Shokrollahi, B. Hassibi, B. M. Hochwald, and W. Sweldens, Reresentation theory for high-rate multile-antenna code design, IEEE Trans. Inform. Theory, vol. 47, , Set.. [] B. Hassibi and B. M. Hochwald, Cayley differential unitary sace-time codes, IEEE Trans. Inform. Theory, vol. 48, , June. [] X.-B. Liang and X.-G. Xia, Unitary signal constellations for differential sace-time modulation with two transmit antennas: Parametric codes, otimal designs, and bounds, IEEE Trans. Inform. Theory, vol. 48,. 9 3, Aug.. [] K. L. Clarkson, W. Sweldens, and A. Zheng, Fast Multile Antenna Differential Decoding, Bell Laboratories, Lucent Technologies, Tech. Re., 999. [3] B. Hassibi and M. Khorrami, Fully-diverse multi-antenna constellations and fixed-oint-free Lie grou, IEEE Trans. Inform. Theory, submitted for ublication. [4] L. Zheng and D. N. C. Tse, Communication on the Grassmann manifold: A geometric aroach to the noncoherent multile-antenna channel, IEEE Trans. Inform. Theory, vol. 48, , Feb.. [5] M. Brehler and M. K. Varanasi, Asymtotic error robability analysis of quadratic receivers in Rayleigh-fading channels with alications to a unified analysis of coherent and noncoherent sace-time receivers, IEEE Trans. Inform. Theory, vol. 47, , Set.. [6] A. Shokrollahi. Design of unitary sace-time codes from reresentation of SU(). [Online]. Available: htt//mars.bell-labs.com [7], A note on double antenna diagonal sace-time codes. [Online]. Available: htt//mars.bell-labs.com [8], Comuting the erformance of unitary sace-time grou constellations from their character table. [Online]. Available: htt//mars.belllabs.com [9] C. Schlegel and A. Grant, Differential sace-time turbo codes, IEEE Trans. Inform. Theory, vol. 49, , Set. 3. [] N. J. A. Sloane.. [Online]. Available: htt:// ~njas/ackings [] T. Hirai, Linear Algebras and Reresentation Theory. Tokyo, Jaan: Sugaku Books,, vol. I, II.