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On groups of complex integers used as QAM signals
On groups of complex integers used as QAM signals Xue-Dong Dong, Cheong Boon Soh *, Erry Gunawan, Li-Zhong Tang
Abstract Recady. the multiplicative Zlij/(l” -t- ?i) w3z used to ohlnin space of 2”’ points and to design that the multiplicative group of to obtain a QAM signal space of The required modulo operation this paper arc able to correct one All rights reserved.
group of units in the a quadrature amplitude ant- error correcting crdes. units in the quotient ring 2”’ ’ points ;md to &sign in this paper i\ simpler and error with m(+rL’ value>.
Gautrian (iudient ring modula;~on (QAM) signal In this paper it is shown Z[i!;(?‘; can Go he used one error currccting codtv. bbck codes constructed in ( 1 IO99 EIsL~ iir Science ‘;I<.
-
---
1. lntroductbn Block they its
are
codes not
quadrature
codes
o\er
the
0020-O~S5/9‘~/Sl9.00 PII:
over suited
finite for
fields
coding
ha\c over
amplitude
n~odttlatic)n
finite
of
fields
!C 1999 Else\icr
the
dcg.ant
lnc
olgebrair
dimensional
(QAM
tiaussian
Science
s 0 0 2 0 -0 2 5 5 ( 9 8 1I 0 0 6 I -0
itrl two
). tlubcr
integer
,411 rishlr
theor>.
signal ring
recrwd.
[ I.21 Z!ij
HoLvever.
cons;tellations
such
considered
linear
modulo
a Gaussian
262
X
-II
Dong
er 01. I Ittfiwttmtion
.ScCncc~.v I13
(1999)
261 -359
prime and the Eisenstein integer ring Z[OJ] = {a + hi 1a, h E Z} modulo a Eisenstein prime, where i = $-ii and w = (-I -I- a)/2 - (- 1 + i&)/2. These codes can be used for QAM signal constellations and can correct more than one error. Egri and Horrigan [3] used Gaussian integers to obtain a 16QAM signal space which is useful for differential coherent detection of a i6QAM signal. Rifi classified multiplicative group of units in the quotient ring Z[i]/(Z* + 29) which gives a QAM signal space of 2’” points, and used these groups to construct nonlinear block codes which citn correct one error in a two-dimensional signal. This error, after quantization, can belong to the set {f 1, k2, fi, f2i, &( I + i), It( I - i)}. In this paper, we factor the multiplicative group of units in the quotient ring Z[i]/(2”) as a direct product of cyclic groups. Then we use these groups to design nonlinear block codes which can correct one error in a two-dimensional signal. This error can take on the values in the set {3”i’(mod 2”) = a,, + b,,ilO < s < 2”-‘.O < t < 4, -2”- ’ < a,,, h,, < 2”-‘ } OI {(I +2i)“i’(mod2”) =a,,+h,,i~O
2. Preliminaries Denote &i by i and a prime integer by p in this paper. The Gaussian integer ring is Z[i] = {u + hi ) a, b E Z; which is the algebraic integer ring of the quadratic number field a(a). The conjugate element of an element x = u + hi is 9 = a - hi. The norm of the element x = a + hi is defined by N(r) = x2* = a2 -I- h’. It is easy to verify that N(rfj) = N(r).N([I). where Y.and b are any elements of Z[i]. The unit group, which consists of all multiplicative inverses of Z[i]. is the cyclic group (i) = (il. &i}. Given r = u + /%/I = c +di #- 0 E Z[i]. let r/~‘//~fi* =x +~i. wherex,p E Q, ;tnd let /j = [x] + [r]i. ;’ = r - /Iit, where for any real number U, [u] denotes hounding to the nearest and smaller integer (e.g. [i] = 0). Then it is easily veri,Ted that r -7 [@I + ;’ and IV(;) < I%‘(/{).Thus, Z[i] ii a Euc!idean ring and therefore has the unique factorization property. Define the funktion r~by o(u) = 1 if &E Z is even and a(u) = -1 if u E Z is odd as in [4].
,t: -I>. nwy
c’t trl. I It!~rwltrrrrkNt
s~~;lvrr~c~s 111 I IVWI
A51 ,‘fiY
263
The irreducible elements in Zjij. up to unitary factors. are: 0 The clement whose norm is a prime intcgcr 1). This norm must take on the value p = 2 or p s I (mod 4). l All rational primes p =: 4k + 3.
3. The multiplicative group of units in the quotient ring Z[i]/(2”) This section is devoted to studying the multiplicative group of units in the quotient ring Z[i]/(F). It will be shown that the group is the cyclic group with order 2 when n = I. and d direct product of two cyclic groups with orders 2’ and 2, respectively. when II = 2 and a direct product of three cyclic groups with orders 711 1.2’1 2 and 22, Itspcctively. ‘1 for a glvcn 11 3 3. Thus these groups give a QA’G
signal
I.. . . elements.
of 2.X. 31. . . . .2”’
Proof. (u) cr=$ (h). Since 2 = (I L i)( 1 - i) := (I + i)‘? ible in Z[i]. we have (0 + hi. I -t i) - I. By Lemma (a) +==3 (c). It is obvious.
and I + i is irreduc(U + hi. 7”) = I c---;$ (n -+ hi. (1 + i)‘i”) = I 1 4 of[4], ((I -+ hi. I -1 iI ~1. I eirr(0)rr(h) = -I. [7
Lemma 3.2. Let II 3 3. Tlwr~.
Proof. It is easy to verify
(I t 3)“ =- I + ?‘( -- I - 3) and ne;ther -1 - 3i nor - 1 - 3i - z is multiple of 1. wher?; is any odd integer. By using induction on n (II > 4). we cab suppose (I - 3)” - = I -+ 2” ‘I?,, 1(iI. bvhere II,, !(i) E Z[i] and neither II,, 1(i) nor II,, i(i) -- : are mul!iple of 2 for ,WIV odd intwerc -,-. Then squaring each side ot’ (I c 3)“’ =: I T 2” ‘It,, ,1 i j UC gt’t (I
+ zi)”
! =: 1
T 3’li,,
,(i) i _+”
it is clear By induction hypothesis. hypothrsis. h,, ,(i) + 2” ‘h,, ,(i)’ - : are multiple tr z 3. WC ha\,e shown that 1.1: 1.1
(I + li)-
‘II,, ,(i)’
~-7 1 -i 2”[/i,,
,(ij -- 2” ‘h,, ,(i)‘].
that neither II,, ,~i J .- 7”~ ‘II,, [ii)’ nor 01‘ 2 Ibr any odd integer z. So when
:
e I + ?‘h,,(i).
where
h,,(i) nor Ir,,(i) - 2 arc mulriplc
II,,(i) E %;ij un~ neither of 2 for any odd integer
z.
(1)
&I
I:
-D. lhrr,~
(*I ul. I Ir!/onmrii~~r
Sc~icvrw
iI.<
I lW.4 I 31
,‘OU
Now Eq. (I) implies that the multiplicative order of I t 2i, mod& visor of 2” ‘. If the multiplicative order of I t ?i. modulo 2”. is 2” 2, the? (I + 2ij” ’ L= 2”k,,(i) $ I for some k,,(i) E Z[i]. ‘But (I $ zi)” z 2” ‘II,, t(i) .t I. It follows that It,, t(i) = X,,(i) which that if,, t(i) are not multiple of 2. Thus the multiplicative order of we can prove that ulo 2”. is 2” ‘. By the same arguments for some odd integers
3” - = Yx,, f I Therefore
the multiplicative
order
2”. is a dia divisor of by Eq. (I), contradicts I + Zi, mod-
x,,.
of 3, modulo
(2) 2”. is 2”~‘.
Cl
Lemma 3.3. Lc*t (x) tkwote
rlrr ~~r*clic .wh~qw~p gcwncrcctcvl hj- x. : ihc cliwct cirrrl Cl thr ir~trrscctiim r~f’gw~p.r. Lcl II > 3. Tlw we* hrrry (tr) (3) n (ij = I. lhl (I + 5) n ((3) . (i)) = I.
prohv
oj’,cyoups
Proof. (a). We shall first prove (3) n (I/ = I. Since any subgroup of a cyclic group is still cyclic, (3) 11 (i) is a cyclic subgroup of (3). Let 3” be a generator of (3) rl (i). where 0 6.j 6 11 - 3. Then there are integers 0
2”).
c jr - 3. then squaring s (-l)“(mod
each side of 3’ z i”(mod
2”). we get
Y).
Since.; ?- I < ti - 2 and the multiplicative be odd. Then we have
order of 3, modulo
2”. is ,” -?, k must
3?” E -l(moJ?‘j. So the multiplicative order 2” :/(Z” ?. 2” ’ ) of 3’ .‘. module 2”. is equa; to the multiplicative order 2 of -1. This results in .j + I := II - 3. But by Eq. (2). 32’ J = 9,, l.y,, I + I for some odd integers s,, 1. Thus we have ,,
lx,, ., -t I z -- I (mod
2”)
This gives I E --I (mod 4). a contradiction. (II). If j = II - 3. then 3”’ ’ :- i”(mod 2” ). So the multiplicative order 2 of 3’” i , modulo 2”. is equal to the multiplicative order Ai(4.k) of i”, Therefore k = 2 and 3” ’ s -I(mod 3”). By the argumenrs of (I) we know that it will result in a contradiction. (III). From the arguments tif (I) and (II) it follows that j = II - 2. Thus we have (3) 0 (i) = (3? ) = I. (b). We shall next prove (1 + 5) r? ( (3:1 . (i!) = I Let ( I -e Zi)” be a generator of the cyclic subgroup (I -J- Zi) r-l (~(3) . {ij ). vvhere 0 <-j < II - 1. Then there are integers 0 <. .S c 2’. ’ and 0
I 3‘
i’(mod
2”).
1: -D.
/hl~
VI 01. I Ir~/i~n~rtrrio~i
.Scitww~
It.3
CIUOUI
31
2OV
265
Suppose, for the sake of contradiction. 0 < ,j i: II - I. We will consider the cases. CM~J I: I = 0. Then (I -t 3)’ z Y(mod 2”). In this case, by comparing the multiplicative orders of (I + 2i)” and k. module 2”. we get #2*-t/ (-pJl) = 2” q,,, 2 .s). This implies s -: 2’ ’ (24 -t- I ) for some integer q. Then we have four
f( 1 + 2i)“j”’
’ ’ E 13” ‘lzY*‘i]‘” E i’”
From
‘(mod
Eqs. (I ) and (3) it follows 1 -I- 2” ‘/I,, , ( i ) 1= I
-t
’ ‘(mod
F). i.e.. (I +. 2i)”
’
2”). that
2” ‘x,, , (mod
2”).
This implies that Z//z,, j(i) - x,, ,. a contradiction to Eq. (I ). 2”). cast 2: I = 2. Then (I -t- 2i)” = -3‘(mod (I). If i <: II -- 2. then (1 -+ 2i)“” = 3zt(nlod 2”). Repeating the arguments of Case I. we will get the desired cy~radiction. rr - 3’ (mod 2”). Squaring each side, we get (II). Ifj =;.I! - 2. then (I t- 2i)z< 3z’(mod 2”). By Lemma 3.2 we get 2” ‘j2.s. So s must has the I = (I t- li)form 2” ‘(2~ + I ) for some integer (I. Then WC’ have
(1 +‘i)~” ~ _ -3”’ “21,’ 1 f -3’” ‘(mod 2”). From
Eqs. (1) and (2). it follows
that
I + 2” ‘/I,, 1(i ) 5 -( I -t 2” lx,, I i( mod 3” 1. This results in I 7 - 1 (mod J), a contradiction. Clrse 3: t = I. Then (I -t 2i)’ --r: 3’ i(mod 2” 1. (I). If i < 11- 2. then (I -F 2i)’ ’ = 3” ’ i’ -= -3”(mod 2”). Repeating the arguments of Case 2, we will get tt;c desired contradiction. (II). Ifj = y - 2. then ( I + 5)” 5: 3‘ . ijmod 2” 1. Squaring each side we get 1 E (I + zi)“’ E -31’(rnod Ztr ). This contradicts (a) of this lemma. CUSP 4: I = 3. Then (I -i- 51’ =” --3’ i(mod 2”). Its proof is himilar to that of Case 3. Thus we must have j ~- II - I and therefore (I +2i)\fi((3)
. (i)j
:: ((I t-3,’
‘) -ZZI.
Theorem 3.1. LP( G,, ckvrot~~I/W t?urftillfic.rrril.~,~~IM~J
-1:
of’mrits iit rfw cpotictrt ring
Z[i!/(2”). T/ICVI ((1) G, = {(~-hi; -- 1” I ,;: t,c 2” ‘.--2’i ’ <: 1~5: 2” '.fl((llrr(/~j E -I}. (IT! IG,,i = 2”: ‘. /CJ GE = (ij.Gz -:: (I -v 2i) f iij.G,, = (I -- 2i) . ‘3) . {i). when II 2 3.
Proof. It is easy to see that
G, :- (a+/>il-2” ’ CU.
Cl
The group G,, can be used as i! 2”’ ’ QAM signal, space. Let So-; {k(lkl)/l<~& the set consisting of (u + hi) being added to every member of the set ,)I,. -1 hen it is easy to verify that G,, = S,’ u S,, u S !, US, ?I S , C-I(SO + (I - i)) (n 3 2). The groups G2 and Gj are illustrated in Figs. I and 2. It is clear that the minimum Euclidean distance between signal points is 6. bvhcn II 2 2.
4. One error-correcting
codes
Now we follow the same ideas as in [4] to construct one error-correcting codes. Consider the set Gl, of all the sequences {gi}i;,. where each s, E G,, and I = (2” i Y = 1 +2i(mod We define
the code C; c G! with
(r”.x’.....rZ” i.e., the scquencc ;‘+ r_ -
?,“):()
< 2” ‘).
the follo\ying
parity
check
‘) (;I,,} ,,,, , E C, if and only
if
1 ;*,,, )I?. n,gI {cl”}
Code C; is :l block code of length 2” ’ over G,,. The transmission rate of the code c’, is R = (3” ’ - I)/?‘. It is easy to verify that the minimum Hamming distance cifr between codewords is 1. So these codes arc maximum distance separable. The miniuxn squared Euclidean distance 11: between uncoded vectors is 2 and between two codewords in code c’, is t/i.: m= 12 (tt > 3). So the asymptotic coding gain G $ IO log,,,( R . ~I;;~‘~/,’ 1 t/R. Thus G =: IO log,,, 6liB =z 7.78 CiB. If we receive any component 0 T. s + .ri of the sequence through an additlvc Gaussian channel. let iC)j = isi -i- [l+li. where for any real number E’. [H] denotes
I 0
tI I
0
*lo
0
0 0
0 0
0
I
0
0 0
0
0
rounding to the nearer and smaller integer as before. se can associate 0 to [Cl). The above process is called quantization. Suppose that, after quantization. there is only one error L !n Z[i] added to the component indexed by k E I. The ret eived sequewe is /: = {/I,},,~., where
?) -1: N,, -t-h,,i / 0 ~3s < 2” ‘.O 6 t < 4. -2”
{ 3’i’(mod
’ i a,,. h,, 5; 3” ‘}. (3)
Proof. Since G,, = (I +- 2i) : (3) . (i) is a direct product of groups. Yi’(mod 2’) = (I,, -i h,,i (0 <.s < 2” ‘.O < f < 4, -2” ’ < u,,. b,,! G 2” ‘), are all different and any single error from the set (3) will give a different syndrome. This completes the proof of the theorem. LI Example
4.1. Let II ‘=L4. Then
l&l
= 27.
I == (2”. x’, . . . .x7). (;*+ i’,l. . . . . ;‘,-) is a codeword
iff
The code C, is able to correct
one error
{i I. f3.15.
ri-7. ii.
i3i.
*5i.
with
the values
in the following
set:
*7i}.
Assume that we transmit the QAM signals (I. 0). (I. -2). (- I. 0). (l.2).(-1.4). (-I.-2) and (-2.3). We let ;‘x! =I. ;*.,: = I-2i, ;‘,I=-I. ;‘,i, =: 1 t- 2j, ;*,, =~ - 1 $- Ji, ;a+ 7: -I - 2i and ;‘Y =: -2 + 3i. Then ;'p
-Ez
- I -(-I
E -7i
(I
t
2i) - (I - 2i)( I f 7i)’
+4i)(l
+-3i)‘+(l
i-2i)(l
-t
I
(I + 2i)’ - (I
-+2i)h-(-2+3i)(l
t
3i)( I i 3i)J +2i)’
(mod 2”).
Now. the eight signals (0. -7). ( I. 0). ( I. -7). (-I. 0). (I. 2). (-I .4). (-I. -2) and (-2.3) will be sent through .sn additive Ciaussian channel. Suppose that. after quantization, one error -3i is added to the third component of the received sequence which will be
X. -D. Dong
et cd. I Infbrtwtion
S&nw
I I.3 : iY9Y)
351 -269
269
The receiver will compute the syndrome s(p) 3 - 7ir0 + z’ + ( I - $)a’ - 2 -t (I + 2i)z’ + (-I + 4i)r5 + (-I - 2i)r“ + (-2 + 3i)x’ z -3ir’ (mod 2’). Thus, there is one error of type -3i in the component indexed by’a2. So there is one error of type (0, -3) in the third component of the received sequence {(0.-7).(1,0),(1,-5),i-1,0).(1.2),(-1,4),(-1,-2),(-2,3)}. Similarly, if cKeuse the index set 1 = {a’ 1 x s 3(mod 2”);06a < 2”-2}, we can construct block codes as nbo~: Nhich are :lble to correct one error with the values in the following set {(I + Zi)‘i’(moJ 2”) =
+ h,,i 10
a,,
5. Conclusion We have classified the multiplicative group of units of Z[i] module a nonprime ideal (2”). These groups can be used to obtain a 2’“-’ QAM signal space and to design block codes for coding QAM signals. The required modulo operation is simpler than !he one in [4]. Nonlinear block codes obtained in this paper are also able to correct one error with more values.
References [I]
K. tluber.
Code>
o\cr
Gaussian
integers.
IEEE
Codes
o\~r
Eknstein
Jacobi
integers,
Tranwctirms
on Inlbrm;ltion
Theory
40 (1994)
207 216. [I]
K. Huber.
Contemporary
Mathematics
168 (I9941
16
179. [3]
R.G. Egri. F.A. Horrigun, A finite group ofcompk mtegers and its application coherent detection oI’QA!vl signals. IEEE Tr,msuctians on lnftvmution Theory 219. [4] J. Rifa. Group3 01 complex integer\ usrd a:, QAM Ggnalh. IEEE Tmnwctwns Theory 41 (1995; 1512. 1517.
IO differentially 40 (1994) 216 on Inlbrmation
Abstract Recady. the multiplicative Zlij/(l” -t- ?i) w3z used to ohlnin space of 2”’ points and to design that the multiplicative group of to obtain a QAM signal space of The required modulo operation this paper arc able to correct one All rights reserved.
group of units in the a quadrature amplitude ant- error correcting crdes. units in the quotient ring 2”’ ’ points ;md to &sign in this paper i\ simpler and error with m(+rL’ value>.
Gautrian (iudient ring modula;~on (QAM) signal In this paper it is shown Z[i!;(?‘; can Go he used one error currccting codtv. bbck codes constructed in ( 1 IO99 EIsL~ iir Science ‘;I<.
-
---
1. lntroductbn Block they its
are
codes not
quadrature
codes
o\er
the
0020-O~S5/9‘~/Sl9.00 PII:
over suited
finite for
fields
coding
ha\c over
amplitude
n~odttlatic)n
finite
of
fields
!C 1999 Else\icr
the
dcg.ant
lnc
olgebrair
dimensional
(QAM
tiaussian
Science
s 0 0 2 0 -0 2 5 5 ( 9 8 1I 0 0 6 I -0
itrl two
). tlubcr
integer
,411 rishlr
theor>.
signal ring
recrwd.
[ I.21 Z!ij
HoLvever.
cons;tellations
such
considered
linear
modulo
a Gaussian
262
X
-II
Dong
er 01. I Ittfiwttmtion
.ScCncc~.v I13
(1999)
261 -359
prime and the Eisenstein integer ring Z[OJ] = {a + hi 1a, h E Z} modulo a Eisenstein prime, where i = $-ii and w = (-I -I- a)/2 - (- 1 + i&)/2. These codes can be used for QAM signal constellations and can correct more than one error. Egri and Horrigan [3] used Gaussian integers to obtain a 16QAM signal space which is useful for differential coherent detection of a i6QAM signal. Rifi classified multiplicative group of units in the quotient ring Z[i]/(Z* + 29) which gives a QAM signal space of 2’” points, and used these groups to construct nonlinear block codes which citn correct one error in a two-dimensional signal. This error, after quantization, can belong to the set {f 1, k2, fi, f2i, &( I + i), It( I - i)}. In this paper, we factor the multiplicative group of units in the quotient ring Z[i]/(2”) as a direct product of cyclic groups. Then we use these groups to design nonlinear block codes which can correct one error in a two-dimensional signal. This error can take on the values in the set {3”i’(mod 2”) = a,, + b,,ilO < s < 2”-‘.O < t < 4, -2”- ’ < a,,, h,, < 2”-‘ } OI {(I +2i)“i’(mod2”) =a,,+h,,i~O
2. Preliminaries Denote &i by i and a prime integer by p in this paper. The Gaussian integer ring is Z[i] = {u + hi ) a, b E Z; which is the algebraic integer ring of the quadratic number field a(a). The conjugate element of an element x = u + hi is 9 = a - hi. The norm of the element x = a + hi is defined by N(r) = x2* = a2 -I- h’. It is easy to verify that N(rfj) = N(r).N([I). where Y.and b are any elements of Z[i]. The unit group, which consists of all multiplicative inverses of Z[i]. is the cyclic group (i) = (il. &i}. Given r = u + /%/I = c +di #- 0 E Z[i]. let r/~‘//~fi* =x +~i. wherex,p E Q, ;tnd let /j = [x] + [r]i. ;’ = r - /Iit, where for any real number U, [u] denotes hounding to the nearest and smaller integer (e.g. [i] = 0). Then it is easily veri,Ted that r -7 [@I + ;’ and IV(;) < I%‘(/{).Thus, Z[i] ii a Euc!idean ring and therefore has the unique factorization property. Define the funktion r~by o(u) = 1 if &E Z is even and a(u) = -1 if u E Z is odd as in [4].
,t: -I>. nwy
c’t trl. I It!~rwltrrrrkNt
s~~;lvrr~c~s 111 I IVWI
A51 ,‘fiY
263
The irreducible elements in Zjij. up to unitary factors. are: 0 The clement whose norm is a prime intcgcr 1). This norm must take on the value p = 2 or p s I (mod 4). l All rational primes p =: 4k + 3.
3. The multiplicative group of units in the quotient ring Z[i]/(2”) This section is devoted to studying the multiplicative group of units in the quotient ring Z[i]/(F). It will be shown that the group is the cyclic group with order 2 when n = I. and d direct product of two cyclic groups with orders 2’ and 2, respectively. when II = 2 and a direct product of three cyclic groups with orders 711 1.2’1 2 and 22, Itspcctively. ‘1 for a glvcn 11 3 3. Thus these groups give a QA’G
signal
I.. . . elements.
of 2.X. 31. . . . .2”’
Proof. (u) cr=$ (h). Since 2 = (I L i)( 1 - i) := (I + i)‘? ible in Z[i]. we have (0 + hi. I -t i) - I. By Lemma (a) +==3 (c). It is obvious.
and I + i is irreduc(U + hi. 7”) = I c---;$ (n -+ hi. (1 + i)‘i”) = I 1 4 of[4], ((I -+ hi. I -1 iI ~1. I eirr(0)rr(h) = -I. [7
Lemma 3.2. Let II 3 3. Tlwr~.
Proof. It is easy to verify
(I t 3)“ =- I + ?‘( -- I - 3) and ne;ther -1 - 3i nor - 1 - 3i - z is multiple of 1. wher?; is any odd integer. By using induction on n (II > 4). we cab suppose (I - 3)” - = I -+ 2” ‘I?,, 1(iI. bvhere II,, !(i) E Z[i] and neither II,, 1(i) nor II,, i(i) -- : are mul!iple of 2 for ,WIV odd intwerc -,-. Then squaring each side ot’ (I c 3)“’ =: I T 2” ‘It,, ,1 i j UC gt’t (I
+ zi)”
! =: 1
T 3’li,,
,(i) i _+”
it is clear By induction hypothesis. hypothrsis. h,, ,(i) + 2” ‘h,, ,(i)’ - : are multiple tr z 3. WC ha\,e shown that 1.1: 1.1
(I + li)-
‘II,, ,(i)’
~-7 1 -i 2”[/i,,
,(ij -- 2” ‘h,, ,(i)‘].
that neither II,, ,~i J .- 7”~ ‘II,, [ii)’ nor 01‘ 2 Ibr any odd integer z. So when
:
e I + ?‘h,,(i).
where
h,,(i) nor Ir,,(i) - 2 arc mulriplc
II,,(i) E %;ij un~ neither of 2 for any odd integer
z.
(1)
&I
I:
-D. lhrr,~
(*I ul. I Ir!/onmrii~~r
Sc~icvrw
iI.<
I lW.4 I 31
,‘OU
Now Eq. (I) implies that the multiplicative order of I t 2i, mod& visor of 2” ‘. If the multiplicative order of I t ?i. modulo 2”. is 2” 2, the? (I + 2ij” ’ L= 2”k,,(i) $ I for some k,,(i) E Z[i]. ‘But (I $ zi)” z 2” ‘II,, t(i) .t I. It follows that It,, t(i) = X,,(i) which that if,, t(i) are not multiple of 2. Thus the multiplicative order of we can prove that ulo 2”. is 2” ‘. By the same arguments for some odd integers
3” - = Yx,, f I Therefore
the multiplicative
order
2”. is a dia divisor of by Eq. (I), contradicts I + Zi, mod-
x,,.
of 3, modulo
(2) 2”. is 2”~‘.
Cl
Lemma 3.3. Lc*t (x) tkwote
rlrr ~~r*clic .wh~qw~p gcwncrcctcvl hj- x. : ihc cliwct cirrrl Cl thr ir~trrscctiim r~f’gw~p.r. Lcl II > 3. Tlw we* hrrry (tr) (3) n (ij = I. lhl (I + 5) n ((3) . (i)) = I.
prohv
oj’,cyoups
Proof. (a). We shall first prove (3) n (I/ = I. Since any subgroup of a cyclic group is still cyclic, (3) 11 (i) is a cyclic subgroup of (3). Let 3” be a generator of (3) rl (i). where 0 6.j 6 11 - 3. Then there are integers 0
2”).
c jr - 3. then squaring s (-l)“(mod
each side of 3’ z i”(mod
2”). we get
Y).
Since.; ?- I < ti - 2 and the multiplicative be odd. Then we have
order of 3, modulo
2”. is ,” -?, k must
3?” E -l(moJ?‘j. So the multiplicative order 2” :/(Z” ?. 2” ’ ) of 3’ .‘. module 2”. is equa; to the multiplicative order 2 of -1. This results in .j + I := II - 3. But by Eq. (2). 32’ J = 9,, l.y,, I + I for some odd integers s,, 1. Thus we have ,,
lx,, ., -t I z -- I (mod
2”)
This gives I E --I (mod 4). a contradiction. (II). If j = II - 3. then 3”’ ’ :- i”(mod 2” ). So the multiplicative order 2 of 3’” i , modulo 2”. is equal to the multiplicative order Ai(4.k) of i”, Therefore k = 2 and 3” ’ s -I(mod 3”). By the argumenrs of (I) we know that it will result in a contradiction. (III). From the arguments tif (I) and (II) it follows that j = II - 2. Thus we have (3) 0 (i) = (3? ) = I. (b). We shall next prove (1 + 5) r? ( (3:1 . (i!) = I Let ( I -e Zi)” be a generator of the cyclic subgroup (I -J- Zi) r-l (~(3) . {ij ). vvhere 0 <-j < II - 1. Then there are integers 0 <. .S c 2’. ’ and 0
I 3‘
i’(mod
2”).
1: -D.
/hl~
VI 01. I Ir~/i~n~rtrrio~i
.Scitww~
It.3
CIUOUI
31
2OV
265
Suppose, for the sake of contradiction. 0 < ,j i: II - I. We will consider the cases. CM~J I: I = 0. Then (I -t 3)’ z Y(mod 2”). In this case, by comparing the multiplicative orders of (I + 2i)” and k. module 2”. we get #2*-t/ (-pJl) = 2” q,,, 2 .s). This implies s -: 2’ ’ (24 -t- I ) for some integer q. Then we have four
f( 1 + 2i)“j”’
’ ’ E 13” ‘lzY*‘i]‘” E i’”
From
‘(mod
Eqs. (I ) and (3) it follows 1 -I- 2” ‘/I,, , ( i ) 1= I
-t
’ ‘(mod
F). i.e.. (I +. 2i)”
’
2”). that
2” ‘x,, , (mod
2”).
This implies that Z//z,, j(i) - x,, ,. a contradiction to Eq. (I ). 2”). cast 2: I = 2. Then (I -t- 2i)” = -3‘(mod (I). If i <: II -- 2. then (1 -+ 2i)“” = 3zt(nlod 2”). Repeating the arguments of Case I. we will get the desired cy~radiction. rr - 3’ (mod 2”). Squaring each side, we get (II). Ifj =;.I! - 2. then (I t- 2i)z< 3z’(mod 2”). By Lemma 3.2 we get 2” ‘j2.s. So s must has the I = (I t- li)form 2” ‘(2~ + I ) for some integer (I. Then WC’ have
(1 +‘i)~” ~ _ -3”’ “21,’ 1 f -3’” ‘(mod 2”). From
Eqs. (1) and (2). it follows
that
I + 2” ‘/I,, 1(i ) 5 -( I -t 2” lx,, I i( mod 3” 1. This results in I 7 - 1 (mod J), a contradiction. Clrse 3: t = I. Then (I -t 2i)’ --r: 3’ i(mod 2” 1. (I). If i < 11- 2. then (I -F 2i)’ ’ = 3” ’ i’ -= -3”(mod 2”). Repeating the arguments of Case 2, we will get tt;c desired contradiction. (II). Ifj = y - 2. then ( I + 5)” 5: 3‘ . ijmod 2” 1. Squaring each side we get 1 E (I + zi)“’ E -31’(rnod Ztr ). This contradicts (a) of this lemma. CUSP 4: I = 3. Then (I -i- 51’ =” --3’ i(mod 2”). Its proof is himilar to that of Case 3. Thus we must have j ~- II - I and therefore (I +2i)\fi((3)
. (i)j
:: ((I t-3,’
‘) -ZZI.
Theorem 3.1. LP( G,, ckvrot~~I/W t?urftillfic.rrril.~,~~IM~J
-1:
of’mrits iit rfw cpotictrt ring
Z[i!/(2”). T/ICVI ((1) G, = {(~-hi; -- 1” I ,;: t,c 2” ‘.--2’i ’ <: 1~5: 2” '.fl((llrr(/~j E -I}. (IT! IG,,i = 2”: ‘. /CJ GE = (ij.Gz -:: (I -v 2i) f iij.G,, = (I -- 2i) . ‘3) . {i). when II 2 3.
Proof. It is easy to see that
G, :- (a+/>il-2” ’ CU.
Cl
The group G,, can be used as i! 2”’ ’ QAM signal, space. Let So-; {k(lkl)/l<~& the set consisting of (u + hi) being added to every member of the set ,)I,. -1 hen it is easy to verify that G,, = S,’ u S,, u S !, US, ?I S , C-I(SO + (I - i)) (n 3 2). The groups G2 and Gj are illustrated in Figs. I and 2. It is clear that the minimum Euclidean distance between signal points is 6. bvhcn II 2 2.
4. One error-correcting
codes
Now we follow the same ideas as in [4] to construct one error-correcting codes. Consider the set Gl, of all the sequences {gi}i;,. where each s, E G,, and I = (2” i Y = 1 +2i(mod We define
the code C; c G! with
(r”.x’.....rZ” i.e., the scquencc ;‘+ r_ -
?,“):()
< 2” ‘).
the follo\ying
parity
check
‘) (;I,,} ,,,, , E C, if and only
if
1 ;*,,, )I?. n,gI {cl”}
Code C; is :l block code of length 2” ’ over G,,. The transmission rate of the code c’, is R = (3” ’ - I)/?‘. It is easy to verify that the minimum Hamming distance cifr between codewords is 1. So these codes arc maximum distance separable. The miniuxn squared Euclidean distance 11: between uncoded vectors is 2 and between two codewords in code c’, is t/i.: m= 12 (tt > 3). So the asymptotic coding gain G $ IO log,,,( R . ~I;;~‘~/,’ 1 t/R. Thus G =: IO log,,, 6liB =z 7.78 CiB. If we receive any component 0 T. s + .ri of the sequence through an additlvc Gaussian channel. let iC)j = isi -i- [l+li. where for any real number E’. [H] denotes
I 0
tI I
0
*lo
0
0 0
0 0
0
I
0
0 0
0
0
rounding to the nearer and smaller integer as before. se can associate 0 to [Cl). The above process is called quantization. Suppose that, after quantization. there is only one error L !n Z[i] added to the component indexed by k E I. The ret eived sequewe is /: = {/I,},,~., where
?) -1: N,, -t-h,,i / 0 ~3s < 2” ‘.O 6 t < 4. -2”
{ 3’i’(mod
’ i a,,. h,, 5; 3” ‘}. (3)
Proof. Since G,, = (I +- 2i) : (3) . (i) is a direct product of groups. Yi’(mod 2’) = (I,, -i h,,i (0 <.s < 2” ‘.O < f < 4, -2” ’ < u,,. b,,! G 2” ‘), are all different and any single error from the set (3) will give a different syndrome. This completes the proof of the theorem. LI Example
4.1. Let II ‘=L4. Then
l&l
= 27.
I == (2”. x’, . . . .x7). (;*+ i’,l. . . . . ;‘,-) is a codeword
iff
The code C, is able to correct
one error
{i I. f3.15.
ri-7. ii.
i3i.
*5i.
with
the values
in the following
set:
*7i}.
Assume that we transmit the QAM signals (I. 0). (I. -2). (- I. 0). (l.2).(-1.4). (-I.-2) and (-2.3). We let ;‘x! =I. ;*.,: = I-2i, ;‘,I=-I. ;‘,i, =: 1 t- 2j, ;*,, =~ - 1 $- Ji, ;a+ 7: -I - 2i and ;‘Y =: -2 + 3i. Then ;'p
-Ez
- I -(-I
E -7i
(I
t
2i) - (I - 2i)( I f 7i)’
+4i)(l
+-3i)‘+(l
i-2i)(l
-t
I
(I + 2i)’ - (I
-+2i)h-(-2+3i)(l
t
3i)( I i 3i)J +2i)’
(mod 2”).
Now. the eight signals (0. -7). ( I. 0). ( I. -7). (-I. 0). (I. 2). (-I .4). (-I. -2) and (-2.3) will be sent through .sn additive Ciaussian channel. Suppose that. after quantization, one error -3i is added to the third component of the received sequence which will be
X. -D. Dong
et cd. I Infbrtwtion
S&nw
I I.3 : iY9Y)
351 -269
269
The receiver will compute the syndrome s(p) 3 - 7ir0 + z’ + ( I - $)a’ - 2 -t (I + 2i)z’ + (-I + 4i)r5 + (-I - 2i)r“ + (-2 + 3i)x’ z -3ir’ (mod 2’). Thus, there is one error of type -3i in the component indexed by’a2. So there is one error of type (0, -3) in the third component of the received sequence {(0.-7).(1,0),(1,-5),i-1,0).(1.2),(-1,4),(-1,-2),(-2,3)}. Similarly, if cKeuse the index set 1 = {a’ 1 x s 3(mod 2”);06a < 2”-2}, we can construct block codes as nbo~: Nhich are :lble to correct one error with the values in the following set {(I + Zi)‘i’(moJ 2”) =
+ h,,i 10
a,,
5. Conclusion We have classified the multiplicative group of units of Z[i] module a nonprime ideal (2”). These groups can be used to obtain a 2’“-’ QAM signal space and to design block codes for coding QAM signals. The required modulo operation is simpler than !he one in [4]. Nonlinear block codes obtained in this paper are also able to correct one error with more values.
References [I]
K. tluber.
Code>
o\cr
Gaussian
integers.
IEEE
Codes
o\~r
Eknstein
Jacobi
integers,
Tranwctirms
on Inlbrm;ltion
Theory
40 (1994)
207 216. [I]
K. Huber.
Contemporary
Mathematics
168 (I9941
16
179. [3]
R.G. Egri. F.A. Horrigun, A finite group ofcompk mtegers and its application coherent detection oI’QA!vl signals. IEEE Tr,msuctians on lnftvmution Theory 219. [4] J. Rifa. Group3 01 complex integer\ usrd a:, QAM Ggnalh. IEEE Tmnwctwns Theory 41 (1995; 1512. 1517.
IO differentially 40 (1994) 216 on Inlbrmation