='''"' >• Thu Hucmg vo Btg Tap ohi KHOA HOC & C6NG NGHE 166(06)207-214
THE UNITS OP THE CHAIN RING /"*• > A N D AN APPLICATION
(" )
Huong T, T. Nguyen", Phuc T, Do, Hoa Q. Nguyen, Hai T, Hoang, Linh M.
T, Tran
College of Economics and Business Admimstratton - TNU
ABSTRACT
The units of the chain ring H4 = ^ f e ^ = F2- •+• u¥2^ + u^¥2^ -b u^F2". are partitioned into 4 distinct types. It is shown that for any unit a of Type k, a unit a' of Type k* can be constructed for all fc ^ 0,1,2,3. The units of H4 of the forma = ao+uai-\-u^a2 + u^ot3, where ao, 0:1,0:2, 0:3 G Fa-n, QQT^O, a i 7^0, are considered in details. As an application, it is shown that self-dual a-constacyclic codes of length 2* over Tli exist. Wc can also prove that it is unique.
Keyivords: Constacyclic codes, cyclic codes, dual codes, chain rings.
1, I N T R O D U C T I O N
The classes of cyclic and negacyclic codes in particular, and constacyclic codes in general, play a very significant role in the theory of errjOr-correcting codes. Let F be a finite field of characteristic p and a be a nonzero element of F. a-constacyclic codes of length n over F are classified as the ideals {g{x)) of the quotient ring F[x]/ (x" - a), where the generator polynomial g{x) is the unique monic polynimial of minimum degree in the code, which is a divisor of x" — ex.
Many well known codes, such as BCH, Kerdock, Golay, Reed-Muller, Preparata, Justesen, and binary Hamming codes, are either cyclic codes or constructed from cyclic codes. Cyclic codes over finite fields were first studied in the late 1950s by Prange [25], while negacyclic codes over finite fields were initiated by Berlekamp in the late 1960s [2].
The case when the code length n is divisible by the characteristic p of the field yields the so-called repeated-root codes, which were first studied since 1967 by Berman [3], and then in the 1970's and 1980's by several authors such as Massey ei al. [19], Falkncr et al. [10].
However, repeated-root codes over finite fields were investigated in the most generality in the 1990's by Castagnoli et al. [5], and van Lint [30], where they showed that repeated-root cyclic codes have a concatenated construction, and are asymptotically bad, Nevertheless, such codes are optimal in a few cases, that motivates researchers to further study this class of codes (see, for example, [21]),
'' The class of finite rings of the form ^^0y - Fpm + uFpm has been used widely as alphabets
Nguyan Thi Thu Huoi^ vd Dtg Tap chi KHOA HQC & CONG NGHE 166(06) 207 - 214 of certain constacyclic codes. For example, the structure of ^ ^ is interesting, because this ring lies between F4 and S4 in the sense that it is additively analogous to F4, and multiplicative!}' analogous to Z4, Codes over - 7 ^ have been extensively studied by many researchers, whose work includes cyclic and self-dual codes, decoding of cyclic codes, Type II codes, duadic codes, repeated-root constacyclic codes.
The rest of this paper is organized as follows. After presenting some preliminary concepts about constacyclic codes over finite commutative rings in Section 2, we classify and inves- tigate thfe units of the ring 7^4 = ^^f = F2m + UF2- + w^Fam -i- u^Fam in Section 3. •K.4 is a chain ring with residue field F2'" that contains precisely (2"* - 1)2^"^ units, namely, ao + uai + 1,1^0:2 + w^QSi where ao, a i , a2, as E F2m, ao / 0- We partition these units into 4 distinct types, and we show that for any unit a of Type k, a unit a* of Type k' can be constructed, where 1 < fc < 3.
2. CONSTACYCLIC CODES OVER F I N I T E COMMUTA- TIVE RINGS
Let ii be a finite commutative ring. An ideal J of R is called principal if it is generated by one element. The following equivalent conditions are well-known for the class of finite commutative chain rings (cf [8, Proposition 2.1]).
Proposition 2.1. For a finite commutative nng R the following conditions are eguivalent:
(i) R IS a local nng and the maximal ideal M of R is principal, (ii) R IS a local principal ideal nng,
{iii) R is a chain ring.
Let z be a fixed generator of the maximal ideal M of a finite commutative chain ring R.
Then z is nilpotent and we denote its nilpotency index by vo. The ideals of R form a chain:
fl = < 2 ° > 2 < ^ ' ) 2 - - - 2 < ^ ° - ' > 2 ( , ^ ' = ' ) = {0).
The following is a well-known fact about finite commutative chain rings (cf, [20]).
Proposition 2.2. Let R be a finite commutative chain ring, tfif/i maximal ideal M = (z), and let tu be the nilpotency of z. Then
(a) For some prime p and positive integers k,l {k > I), \R\ = j^,\R\ = p*, and the characteristic of R and R are powers ofp,
(b) Fori ^ 0,1,...,VJ, \{z')\ = \R\^-\ In particular, \R\ - \R\^, i.e., k = lw.
Nguyt:, Thi Thu Huong vaBrg Tgp chi KHOA HOC & C 6 N G NGHE 166(06) 207-214 Given n-tuples x = (xo,xi,.. .,x„_i),y - (yo,yu-.. ,yn-i) G R", their inner product or dot product is defined in the usual way:
X • y = XoS/o + 3;iyi -i -f- Xn-iyn-i,
evaluated in R. x, y are called orthogonal if x • y ^ 0. For a hnear code C over R, its dual code C is the set of rz-tuples over R that are orthogonal to all codewords of G, i.e.,
C-^ = {x\ x - y = 0,Vi/GC}.
A code C is called self-orihogonal if C C C-^, and it is called self-dual if C = C-*-. The following result is well known (cf [8, 14, 23]).
Proposition 2.3. Let R be a finite chain ring of size p°'. The number of codewords in any linear code C of length n over R is p*, for some integer fc, 0 < fc < an. Moreover, the dual code C-'- hasp°'^~'^ codewords, so that \C\ • |C^| = \R\".
Given an n-tuple ( x o , x i , . . . ,x„-i) e R^, the cyclic shift T and negashift u on -R" are defined as usual, i.e.,
T ( X O , X I , . . . , X „ _ I ) ^ (a;„_i,xo,xi,-'- ,x„_2), and
f ( x o , X i , . . . , X n _ i ) ^ {-Xn-l,Xo,Xi, -• • , X„_2).
A code C is called cyclic if T{C) = C, and C is called negacyclic if i/[C) ^ C. More generally, if a is a unit of the ring R, then the a-constacyclic (a-twisted) shift Ta on R^ is the shift
Ta{XQ,X\ ,Xn-l) = (aXn-i,a;o,xi,-- • ,X„_2),
and a code C is said to be cx-constacyclic if ra{C) = C, i.e., if C is closed under the Q-constacyclic shift TQ.
Each codeword c = (co,ci,... ,Cn_i) is customarily identified with its polynomial repre- sentation c(x) — Co + cix ~\- • •. -\- CTI-IX"-~^, and the code C is in turn identified with the set of all polynomial representations of its codewords. Then in the ring . - _U; xc(x) cor- responds to a a-constacyclic shift of c(x). Prom that, the following fact is well-known and straightforward (cf [18]) :
Proposition 2.4. A hnear code C of length n is a-constacyclic over R if and only if G is an ideal of •r-h_) (Hence, this quotient ring is referred to as the ambient nng of the code C).
The dual of a cyclic code is a cyclic code, and the dual of a negacyclic code is a negacyclic code. In general, wc have the following implication of the dual of a a-constacyclic code.
Proposition 2.5. The dual of a a-constacyclic code is a a~^ -constacyclic code.
Nguy§n Ihl Thu Hucmg vdDtg T^p chi KHOA HOC & CONG NGHE 166(06)' 207 - 214 "
The following result is also well known (cf. [7]),
Proposition 2.6. Let R be a finite commutative nng, a be a unit of R and a(x) ^ ao + flix H h a^-ia:'^"', 6(x) ^ 6o + feix -1 h 6n_ix"~-^ G R[x].
Thena{x)b{x) — 0 m /^n_L zf end oniy 2/(00.01,-. -,a„-i) is orthogonal to (bn-i.&n-2,...,&o) and all its a~^ -constacyclic shifts.
For a nonempty subset 5 of the ring fi, the annihilator of 5, denoted by ann{S), is the set ann{S) = {f\fg = 0, for all 5 G 5 } .
Then anri{S) is an ideal of R.
Customarily, for a polynomial / of degree fc, its reciprocal polynomial x'^/(x~^) will be denoted by /*, For example, if
/(x) = 00 + OiX -I h Ofc-iX*"^ -I- OfcX*, then
/*(x) - x''(ao + aix~^ + . . . + ofc-ix"'*"*^ -I- OfeX^*) ^ o^ -I- Ok-ix -f- • • • + aix*^"^ + ooa:*.
Note that (/*)* — / if and only if the constant term of / is nonzero, if and only if deg(/) = deg{/*).
3, THE RING K4 = ^0f- AND ITS UNITS
The ring 7^4 = -^) is a local ring with maximal ideal uTZ^ = (li)^^ . Applying Proposition 2.1, Tli is a chain ring. Then 7^4 can be viewed as 7^.4 ^ F2'n -\-u¥2"^-i-u^F2^ -|-u^F2m. It is closed under 2'"-ary polynomial addition and multiplication modulo u"*. The set ?^4\(tt)K^
is the set of all units of 7^4, it consists of elements of the form ao + ttai + ii^F2™ + u^az,
where ao,ai,02,03 E F2m, ao?^0. More precisely, we have the following result:
Proposition 3.1. Let Tli - ^0^ = F2m + liFam -|- U^JF^^ 4. ^ajp^™. Then (i) 7^4 is a chain ring with maximal ideal {u)j^.
(ii) The ideals ofR^ are {'0^)TI = U^'RA, each ideal (ti*)jj contains 2"'(^~') elements, 0 < j < 4.
Ngu\ ..> Thi Thu Huong wi Dtg Tgp chi KHOA HOC & CONG NGHE 166(06): 207 - 214 {iii) 7^4 has (2"* - 1)2^"* units, they are of the form
ao + u a i + y^0i2 + u^as, where ao, 01,02,03 G ^2'", OOT^O.
For a nonzero code C, let ic denote the smallest integer such that there is a nonzero component of a codeword of C belonging to {u^'^)-jz \ (w''^"'"^)^ • Clearly, 0 < i c < 3, and C C ( ^ ' » > ^ C K ; .
It is known that, over a finite field F, a code C of length n is o- and ^-constacyclic, for two different units a, /? G F , if and only if C ^ {0} or C = F " . Over a fmite ring R, there are many codes satisfying this property. For example, let / be an ideal of R. then / " is a Q-constacyclic code of length -n over R for any unit a of R. In the following, we give some more results for constacyclic codes over the chain ring 7^4.
Proposition 3.2. Let a be a unit oflZ^. If a code G of length n is a-constacyclic over'R.4 thenC is also T-constacyclic for any unitV such thatT—a E {'^^)TI^I /"'" every j > A—ic- Proof Since F - o e {w^)^^ ^ ("^~"^)724' ^^^^^ ^^ ^ element Q E Tl^ such that F = a + u^'^'^C- Consider an arbitrary codeword C of G, by definition of ic, it has the form C = (u'-^co. u'-^ci...., u"^Cn-i). Clearly,
Fu'^cn-i - (a + H''-'<^)U^'^C„_I = au^'^Cn^i.
Thus, G is also a F-constacyclic code. D
Proposition 3.3. Let C be a code of length n overTZi, and a, a' be units ofTZ^ such that Ct- Oi E {U^)T, \ (UJ"'"^)T5 ,^<j<A-icIfG is both a- and a'-constacyclic over TZ^
then {u^'^^^)Z ^ ^- -^^ particular, if a- a' is a unit, then C = ("*^)K4-
Proof Since a - o' 6 {n^}Tz \ {'^^'^^)-R^' ^^^^^ '^ ^ ^^^^ C G 7^4 such that o - o' = u^C- Without loss of generality, we can assume that (co,..., Cn-i) G G where Cn-i ^ u^'^v, for a unit V G Tli. It follows that both (oc„-i, co, • •. ,c„-i) and (a'cn-i, co,. • - ,c,i-i) belong to C, and hence, their difference is in C. Clearly,
(oc„_i,Co,...,Cn-i) - (a'cn-u CQ, • • -,^-i) = ((« - a')c„^i,0,...,0)
= u^+*^Qv{l,0,...,0), so u-'+*<^(l,0,...,0) G G. That means uJ+'^(l, 0,... ,0) and all its cyclic shifts are in C, and hence, {u^'^'^'}^^ C G. In the case that a - o' is a unit, then j = 0. Therefore, (tx'c)^^ CCQ {u'-Tn^, i.e.. C = {u^c)-^. D
We now give a partition for the units of 7^.4 into 4 distinct types. For an integer/c G {1,2,3}, we call a unit o - Qo + uoi -1- ^^02 + u^oa of 7^4 to be of Type fc, if fc is the smallest index
Nguyen Thi Thu Huong va Dtg Tap chi KHOA HOC & C O N G N G H E 166(06): 207 - 214
such t h a t Oft ^ 0. If, in a d d i t i o n , o o = 1, t h e n 1 + uai + u^02 + u ^ 0 3 is said t o b e of T y p e / : ' . If Q, — 0 for all 1 < i < 3 , i.e., t h e u n i t is of t h e form o — a o G F2m, we say t h a t o is of T y p e 0 (or T y p e OMf a o ^ 1). Clearly, 7^4 h a s 2"^ - 1 u n i t s of T y p e 0, a n d (2"* - 1 )22'"(3-fc) u n i t s of T y p e h, which give 2 " ' — 1 T y p e 0 constacyclic codes a n d (2"^ — l)22"^(^~'^) T y p e fc constacyclic codes.
For 1 < fc < 3 , let o b e a u n i t of T y p e fc of 72-4, i.e., a = Oo + u'^Ofc -H u^02 + ti^03,
w h e r e 0 0 , 0 ^ , 0 2 , 0 3 G F 2 ' " , 00 7^ 0, Ofc^O. L e t a ^ 1 -|- u'^Ofc -i- • • • -I- u^a^, where, for fc < z < 3 , Oj — OjO^^ G Fam. T h e n o is a u n i t of T y p e fc*, a n d a = Ooa. Clearly, in the case of a is a u n i t of T y p s 0 a n d o is of T y p e 0*, we also h a v e o = o o o . T h e following t h e o r e m shows t h a t ( w ^ ) ^ is t h e u n i q u e self-dual a - c o n s t a c y c l i c code of l e n g t h n over 7^4.
T h e o r e m 3 . 4 . Let o — 00 -\-uai -\- w^a2 -f-1x^03 be a unit ofTZ^ such that O Q T ^ I . Then (li^)!^ is the unigue self-dual a-constacyclic code of length n over 0.4.
Proof. Since OQ 7^ 1, OQ 7^ Og ^, a — o~^ is a u n i t of 72-4, L e t C b e a self-dual o-constacyciic code of l e n g t h rt. T h e n b y P r o p o s i t i o n 2.5, C is b o t h a - a n d a~-^-constacylic codes. T h u s , P r o p o s i t i o n 3.3 implies t h a t C = {u"^)^^- In Hght of P r o p o s i t i o n 2.7, C-^ = ( M ° ~ * ' ^ ) J ^ , a n d hence, a = 2ic, as required. D
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P H A N T U K H A N G H I C H C U A V A N H ^^f- V A M O T U N O D U N G N g u y i n T h i T h u Huf&ng, D 6 T h a n h P h i i c * , N g u y i n Q u J ' n h H o a , H o a n g T h a n h H a i , T r l n T h j M a i L i n h
Dg,i hoc Kinh Ti va QTKD
P h i n tli kha nghich ciia vanh Tii duoc chia thanh 4 loai phan bigt. Ph&n tii kha nghich cua vanh 7^4 dU(?c xem xet mot each chi t i l t . Nhu mpt ung dyng m a t\f d6i ng§.u cua raa a-constacyclic co do dai 2* trgn Tt^. Chiing t6i c6 thg chdng minh rang no la duy nhat.
K e y w o r d s a n d p h r a s e s : ma constacyclic, ma, md ddi ng&u, vanh chudi
Ngay nhan bdi: 08/02/2017; Ngdy phdn biin: 30/3/2017; Ngdy duy^t ddng: 31/5/2017 '**Te!: 0949374386; Email: [email protected]
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