A Generalization of Quasi-twisted Codes: Multi-twisted codes

0.09 MB·

by  Nuh Aydin

A Generalization of Quasi-twisted Codes:
Multi-twisted codes
NuhAydina, AjdinHalilovic´b
aKenyonCollege,DepartmentofMathematics,Gambier,OH43022
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1 bLumina-TheUniversityofSouth-EastEurope,S,os. Colentina64b,021187Bucharest,Romania
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2

n
a
J Abstract

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Cyclic codes and theirvarious generalizations, such as quasi-twisted(QT) codes,

]
havea special placein algebraic coding theory. Among otherthings,manyof the
T
I best-knownoroptimalcodes havebeen obtained from theseclasses. In this work
.
s we introduce a new generalization of QT codes that we call multi-twisted (MT)
c
[ codes and study some of their basic properties. Presenting several methods of

1 constructing codes in this class and obtaining bounds on the minimum distances,
v we show that there exist codes with good parameters in this class that cannot be
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obtained as QT or constacyclic codes. This suggests that considering this larger
4
0 class in computer searches is promising for constructing codes with better para-
1
meters than currently best-known linear codes. Working with this new class of
0
. codesmotivatedustoconsideraproblemaboutbinomialsoverfinitefieldsandto
1
0 discoveraresult thatisinterestinginitsownright.
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1 Keywords: constacycliccodes, quasi-twistedcodes,best-knowncodes
:
v 2000MSC:94B15, 94B60,94B65
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r
a 1. Introduction andMotivation
Every linear code over a finite field F has three basic parameters: the length
q
(n),thedimension(k),andtheminimumdistance(d)thatdeterminethequalityof
thecode. Oneofthemostimportantandchallengingproblemsofcodingtheoryis
adiscreteoptimizationproblem: determinetheoptimalvaluesoftheseparameters
andconstructcodeswhoseparametersattaintheoptimalvalues. Thisoptimization
problem is very difficult. In general, it is only solved for the cases where either k
orn−kissmall. Thereisadatabaseofbestknownlinearcodeswithupperbounds
onminimumdistancesthatisavailableonline[1]. Thedatabaseisupdatedasnew
codes arediscoveredandreported by researchers.
PreprintsubmittedtoFiniteFieldsandTheirApplications 5ianuarie2017
Computersareoftenusedinsearchingforcodeswithbestparametersbutthere
is an inherent difficulty: computing the minimum distance of a linear code is
computationally intractable (NP-hard) [11]. Since it is not possible to conduct
exhaustive searches for linear codes if the dimension is large, researchers often
focus on promising subclasses of linear codes with rich mathematical structures.
A promisingand fruitful approach has been to focus ontheclass ofquasi-twisted
(QT) codes which includes cyclic, constacyclic, and quasi-cyclic (QC) codes as
special cases. This class of codes is known to contain many codes with good
parameters. In the last few decades, a large number of record-breaking QC and
QT codes havebeen constructed (e.g. [2]-[10]). The search algorithm introduced
in[4]has been highlyeffectiveand used inseveralsubsequentworks([5]-[10]).
Inthiswork,weintroduceanewgeneralizationofQTcodesthatwecallmulti-
twisted (MT) codes. It turns out that this class also generalizes more recently
introduced classes of double cyclic codes ([2, 3]), QCT codes ([15]), and GQC
codes ([14]). After deriving some of their algebraic properties and obtaining a
lowerboundontheminimumdistance,weshowthatfromthisclasswecanobtain
linearcodeswithbest-knownoroptimalparameters thatcannot beobtainedfrom
thesmallerclassesofconstacyclicorQT codes.
Beforeintroducingthisnewclassofcodes,werecallsomefundamentalresults
aboutconstacyclicand QTcodes thatwillbeneeded later.
2. Constacyclicand Quasi-twistedCodes
Constacyclic codes are very well-known in algebraic coding theory. Let a ∈
F∗=F \{0}. AlinearcodeCoverafinitefieldF iscalledconstacyclicwithshift
q q q
constantaifitisclosedundertheconstacyclicshift,i.e. forany(c ,c ,...,c )∈
0 1 n−1
C, T (c ,c ,...,c ):=(c ,c ,c ,...,c )∈C. When a=1, we obtain the
a 0 1 n−1 n−1 0 1 n−2
veryimportantspecialcaseofcycliccodes. Manywell-knowncodesareinstances
ofcycliccodes.
Under the usual isomorphism p : Fn → F [x]/hxn−ai, where p (u) = u +
q q 0
u x+u x2+...+u xn−1, for u=(u ,u ,...,u )∈Fn, it is well-knownthat
1 2 n−1 0 1 n−1 q
a constacyclic code is an ideal in the ring F [x]/hxn−ai. Moreover, for every
q
constacyclic code C there is a unique, monic polynomial of least degree in C
that generates C, i.e. C = hg(x)i= {f(x)g(x) modxn−a : f(x) ∈ F [x]}. This
q
standardgeneratorisadivisorofxn−a,sothatxn−a=g(x)h(x),forsomeh(x)∈
F [x]whichiscalledthecheckpolynomialofC. Notethatthesetofallcodewords
q
canbedescribedasC={f(x)g(x): f(x)∈F [x] and deg(f(x))<deg(h(x))},i.e.
q
theset {g(x),xg(x),x2g(x),...,xk−1g(x)}isabasis forC where k=deg(h(x)).
2
Aconstacycliccodehasmanyothergeneratorsaswellandwehaveacomplete
characterization ofthemas follows.
Lemma 2.1. [4] LetC=hg(x)ibeaconstacycliccodeof lengthnand shiftcon-
stantawithcanonicalgeneratorg(x)andcheckpolynomialh(x)sothatxn−a=
g(x)h(x). Then C = hg′(x)i if and only if g′(x) is of the form g(x)p(x) with
gcd(p(x),h(x))=1.
A linear code C is said to be ℓ-quasi-twisted (ℓ-QT) if, for a positive in-
teger ℓ, it is invariant under Tℓ, that is, whenever (c ,c ,...,c ) ∈ C, then
a 0 1 n−1
(ac ,...,ac ,c ,c ,...,c )∈C aswell. Itisimportanttonotethatwhen
n−ℓ n−1 0 1 n−ℓ−1
gcd(ℓ,n)= 1 we obtain constacyclic codes. We therefore assume that ℓ | n, the
caseℓ=1 correspondingtoconstacycliccodes.
It is well known that algebraically a QT code C of length n = mℓ with shift
constant a is an R-submodule of Rℓ where R=F [x]/hxm−ai. IfC has a single
q
generator of the form (f (x), f (x),..., f (x)) then it is called a 1-generator QT
1 2 ℓ
code, otherwise a multi-generator QT code. Most of the literature on QT codes
focuseson the1-generatorcase. Wewilldo thesameinthiswork.
A certain type of 1-generator QT codes, sometimes called degenerate QT co-
des, is particularly useful and promising when searching for new linear codes.
This is due to the lower bound on the minimum distance given in the following
theorem.
Theorem 2.2 (see[4]). LetC be a 1-generator ℓ-QT code of length n=mℓ with
a generatoroftheform:
(f (x)g(x), f (x)g(x),···, f (x)g(x)), (1)
1 2 ℓ
where g(x), f (x) ∈ F [x]/hxm−ai, such that xm−a = g(x)h(x) and f (x) is re-
i q i
latively prime to h(x) for each i = 1,...,ℓ. Then C is an [n,k,d′] -code where
q
k=m−deg(g(x)),d′ ≥ℓ·d(C ),andd(C )denotestheminimumdistanceofthe
g g
constacycliccodeof lengthmgeneratedbyg(x).
A proof of this theorem is given in [4]. In reality, the actual minimum dis-
tanceofCisoftenconsiderablylargerthanthelowerboundgivenbythetheorem.
Researchers designed algorithms and conducted computer searches based on this
theoremand discoveredmanynew linearcodes ([5]-[10]).
3
3. Multi-twisted Codes
We propose an even more general class of linear codes, which we call multi-
twisted (MT)codes. A multi-twistedmoduleV isan F [x]-moduleoftheform
q
ℓ
V =(cid:213) F [x]/hxmi−a i,
q i
i=1
where a ∈F \{0} and m are (possibly distinct) positiveintegers. An MT code
i q i
is an F [x]-submodule of a multi-twisted moduleV. Equivalently, we can define
q
anMTcodeintermsoftheshiftofacodeword. Namely,alinearcodeC ismulti-
twistedifforany codeword
~c=(c ,...,c ;c ,...,c ;...;c ,...,c )∈C,
1,0 1,m1−1 2,0 2,m2−1 ℓ,0 ℓ,mℓ−1
itsmulti-twistedshift
(a c ,c ,...,c ;a c ,c ,...,c ;...;a c ,...,c )
1 1,m1−1 1,0 1,m1−2 2 2,m2−1 2,0 2,m2−2 ℓ ℓ,mℓ−1 ℓ,mℓ−2
is also a codeword. If we identify a vector~c withC(x)=(c (x),c (x),...,c (x))
1 2 ℓ
where c(x) = c +c x+···+c xmi−1, then the MT shift corresponds to
i i,0 i,1 i,mi−1
xC(x)=(xc1(x) modxm1−a1,...,xcℓ(x) modxmℓ−aℓ).
Note that cyclic, constacyclic, QC, and QT codes are all (permutation equi-
valent to) special cases of MT codes. For example, QT codes are obtained as a
special case when m =m =···=m and a =a =···=a . Moreover, more
1 2 ℓ 1 2 ℓ
recently introducedclassesofcodes calledgeneralized quasi-cyclic(GQC) codes
[14], QCT codes [15], and double cyclic codes [12, 13] can be viewed as special
cases ofMTcodes.
An MT code is a one-generator code if it is generated by a single element of
V. Thiswork willfocusprimarilyon one-generatorMTcodes.
Our next goal about MT codes is to find a lower bound on the minimum dis-
tance of a 1-generator MT code similar to the one in Theorem 2.2. This leads to
considering the greatest common divisor of two binomials xn1−a and xn2−a .
1 2
Working on this question, we discovered a result about the greatest common di-
visor of two binomials xn1−a and xn2 −a over F which we believe is a new
1 2 q
result about polynomials over finite fields, and interesting in its own right. We
stateand provethisresultinthenextsection.
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4. A ResultAbout BinomialsoverFiniteFields
ConsideringgeneratorsofMTcodesbringsuptheproblemofdeterminingthe
greatest common divisor of two binomials of the form xn1−a and xn2−a . We
1 2
foundthat thegcd oftwo suchpolynomialsiseither1, oranotherbinomialofthe
sameform. Theprecisestatementandaproofare as follows.
Theorem 4.1. LetP={f ∈F [x]: f =xm−a witha∈F∗,m∈N,gcd(m,q)=1}.
q q
Thenfor f,g∈P,gcd(f,g)iseither1oroftheformxgcd(deg(f),deg(g))−a,forsome
a∈F . In particular,gcd(f,g)∈P∪{1}.
q
PROOF. We will let ordm(q) denote the multiplicative order of q modm, i.e. it
isthesmallestpositiveintegerk suchthatqk ≡1 modm. Foranon-zeroelement
a ∈F∗, |a| denotes the order of a in the multiplicative group of (F∗,·). Let f =
q q
xn1−a1 and g=xn2−a2 for some n1,n2 ∈N and non-zero elements a1,a2 ∈Fq
with r =|a | and r =|a |. Let s =ord (q) and s =ord (q). It is known
1 1 2 2 1 n r 2 n r
1 1 2 2
that f andgsplitintolinearfactorsintheextensionsFqs1 andFqs2,respectively. In
order to find a common extensionof these two fields, considers=ord (q).
n r n r
1 1 2 2
We claim that Fqs1,Fqs2 ⊆Fqs, that is, s1|s and s2|s. Indeed, since n1r1n2r2|qs−1,
we have, in particular, n r |qs−1. Since s = ord (q), it follows that s |s.
1 1 1 n r 1
1 1
Similarlyweobtains |s.
2
Letz ∈Fqs beaprimitiven1n2throotofunity. Thenz n2 andz n1 areprimitive
n th andn throotsofunity,respectively.
1 2
If f and g do not have a common root, then gcd(f,g)=1, and we are done.
Suppose now that there exists a common root d of f and g, that is, d is an n th
1
rootofa and n th rootofa . It followsthat therootsof f are
1 2 2
d ,dz n2,d (z n2)2,...,d (z n2)n1−2 and d (z n2)n1−1,
and therootsofg are
d ,dz n1,d (z n1)2,...,d (z n1)n2−2 and d (z n1)n2−1.
Thesetofrootsofgcd(f,g)istheintersectionofthetwosetsabove. Notethat
thesesets areactually cosetsofthemultiplicativesubgroups
{1,z n2,(z n2)2,...,(z n2)n1−1} and {1,z n1,(z n1)2,...,(z n1)n2−1} of Fqs, respecti-
vely. It follows that the set of roots of gcd(f,g) is a coset of the intersection of
these subgroups. Therefore, z n1n2/d is a generator (primitive element) of the su-
bgroup of the intersection where d = gcd(n ,n ). Hence, the roots of gcd(f,g)
1 2
5
are
d ,dz n1n2/d,d (z n1n2/d)2,...,d (z n1n2/d)d−1.
Therefore,deg(gcd(f,g))=gcd(n ,n )=gcd(deg(f),deg(g)). Finally,weshow
1 2
that a := d d ∈ F to prove that gcd(f,g) is of the desired form xm−a, hence
q
completing the proof. Write n = dt and n = dt for some relatively prime
1 1 2 2
integerst ,t . Sincegcd(t ,t )=1,thereexistintegersu,vsuchthatut +vt =1.
1 2 1 2 1 2
Weknowthatd dt1 =a and d dt2 =a , whichimplies
1 2
a=d d =d udt1+vdt2 =au·av ∈F .
1 2 q
Example4.2. Over F , gcd(x10−4,x15−1) = x5−2, gcd(x11−5,x16−4) =
7
x−3 andgcd(x12−3,x15−4)=1.
This theorem has some implicationsfor constacycliccodes. Suppose a ,a ∈
1 2
Fq and n1,n2 ∈ Z+ are such that gcd(xn1−a1,xn2−a2) = xm−a for some non-
zero m∈Z and a∈F . Any constacyclic codeC of length m with shift constant
q
a has a generator g(x) that divides xm−a, hence g(x)|(xn1−a1) and g(x)|(xn2−
a ). Therefore, we observe that the polynomial g(x) can also be regarded as the
2
(standard) generator of a constacyclic codeC of length n with shift constant a
1 1 1
as well as thegenerator of a constacycliccodeC of length n with shift constant
2 2
a .
2
5. More onConstructions ofMT Codes andTheir Parameters
We frequently have gcd(xn1−a1,xn2 −a2) = 1. Consider a 1-generator MT
codeC in this case with a generator of the form hg (x),g (x)i where g (x)|xn1−
1 2 1
a and g (x)|xn2 −a . Clearly, gcd(g (x),g (x)) = 1 as well. If C = hg (x)i
1 2 2 1 2 1 1
with parameters [n ,k ,d ] andC =hg (x)i with parameters [n ,k ,d ], then we
1 1 1 2 2 2 2 2
observe thatC has parameters [n +n ,k +k ,min{d ,d }]. We formally prove
1 2 1 2 1 2
thisinthenexttheorem.
Theorem 5.1. Letn1,n2∈Z+,a1,a2∈F∗qbesuchthatgcd(xn1−a1,xn2−a2)=1.
Let xn1 −a = g (x)h (x) and xn2 −a = g (x)h (x) with k = deg(h (x)),k =
1 1 1 2 2 2 1 1 2
deg(h (x)). Let C = hg (x)i and C = hg (x)i with parameters [n ,k ,d ] and
2 1 1 2 2 1 1 1
[n ,k ,d ], respectively. Then the MT code C with generator hg (x),g (x)i has
2 2 2 1 2
parameters[n,k,d],where n=n +n , k=k +k , andd =min{d ,d }.
1 2 1 2 1 2
PROOF. The assertions on length of the MT codeC is clear. To prove the asser-
tion on dimension, we show that the set S = {xi·(g (x),g (x)): 0 ≤ i ≤ k−1},
1 2
6
wherek=k +k ,isabasisforC. Supposethat f(x)·(g (x),g (x))=(f(x)g (x)
1 2 1 2 1
modxn1 −a , f(x)g (x) modxn2 −a ) = 0. Then f(x)g (x) ≡ 0 modxn1 −a
1 2 2 1 1
and f(x)g2(x) ≡ 0 modxn2 −a2. Therefore, h1(x)|f(x) and h2(x)|f(x). Since
h (x)andh (x)arerelativelyprime(becausegcd(xn1−a ,xn2−a )=1),wehave
1 2 1 2
h (x)h (x)|f(x), and hence deg(f(x))≥k. This implies the linear independence
1 2
ofvectorsinS.
It remains to show that S is a set of generators for C. For this it suffices
to show that for every f(x) ∈ F [x] with deg(f(x)) ≥ k there exists r(x) ∈ F [x]
q q
with deg(r(x)) < k such that f(x)·(g (x),g (x)) = r(x)·(g (x),g (x)). So let
1 2 1 2
f(x) ∈ F [x] be an arbitrary polynomial with deg(f(x)) ≥ k. We can write f(x)
q
as f(x) = h (x)h (x)q(x)+r(x), for some q(x),r(x) ∈ F [x] where deg(r(x)) <
1 2 q
deg(h (x)h (x))=k. It followsthat
1 2
f(x)·(g (x),g (x))=(f(x)g (x) modxn1−a , f(x)g (x) modxn2−a )
1 2 1 1 2 2
=(r(x)g (x) modxn1−a ,r(x)g (x) modxn2−a )
1 1 2 2
=r(x)·(g (x),g (x)).
1 2
Finally, to show that d = min{d ,d }, let u(x) =t(x)g (x) ∈C be a codeword
1 2 1 1
of minimum weight inC . Since gcd(h (x),h (x))=1, we haveC =hg (x)i=
1 1 2 1 1
hh (x)g (x)i. Hence, u(x) = t′(x)h (x)g (x) for some t′(x) ∈ F [x] (degree of
2 1 2 1 q
which can be taken to be < k ). Letting f(x) = t′(x)h (x) and considering the
1 2
codeword f(x)·(g (x),g (x)), we find a codeword (u(x),0)∈C of weight d in
1 2 1
C. Wecan similarlyshowthatthereexistsacodeword ofweightd inC as well.
2
Given this result, we cannot hope to find codes with good parameters under
the conditions of the above proposition. We need to look for alternative ways to
construct MT codes with potentially high minimum distances. Before coming to
more promising constructions, let us first disqualify another case where no MT
codes withhighminimumdistancescan beexpected.
Theorem 5.2. Leta ,a ∈F∗,n 6=n ∈Z+besuchthatgcd(xn1−a ,xn2−a )=
1 2 q 1 2 1 2
xm−a=g(x)h(x),forsomem∈Z+,a∈Fq andletxn1−a1 =g(x)h(x)h1(x)and
xn2 −a = g(x)h(x)h (x) with k = deg(h(x)),k = deg(h (x)), k = deg(h (x)).
2 2 1 1 2 2
Let C be the MT code of length n +n with generator hg(x)p (x),g(x)p (x)i,
1 2 1 2
where gcd(h(x)h (x),p (x))=1 and gcd(h(x)h (x),p (x))=1. ThenC has pa-
1 1 2 2
rameters[n +n ,k+k +k ,d]with d ≤2.
1 2 1 2
PROOF. An argument identicalto theoneused in Proposition5.1 showsthat S=
7
{xi·(g(x)p (x),g(x)p (x)) : 0 ≤i ≤ k+k +k −1} is a basis forC, and hence
1 2 1 2
provestheassertionaboutthedimension.
In order to see that minimum distance is at most 2, we first assume, w.l.o.g.,
that n > n and observe thatC is also generated by hg(x),g(x)p (x)i. Then we
1 2 2
notethath(x)h (x)·(g(x),g(x)p (x))=(xn2−a ,0),whichisacodewordofwei-
2 2 2
ght2.
Note that the theorem above points out a significant difference between QT
and MT codes, since many good codes from the class of QT codes have genera-
tors of the form hg(x),g(x)p(x)i, where gcd(h(x),p(x))=1 [4]. Hence, we need
to consider MT codes with generators of different form to look for codes with
potentially high minimum distances. The next theorem presents such a subclass
ofMT codes. Afterwards we willintroduceaclass ofpromisingsubcodes ofMT
codes.
Theorem 5.3. Let a1,a2 ∈F∗q, n1,n2 ∈Z+ be such that gcd(xn1−a1,xn2−a2)=
xm−a=g(x)h(x),forsomem∈Z+,a∈Fq andletxn1−a1 =g(x)h(x)h1(x)and
xn2 −a = g(x)h(x)h (x) with k = deg(h(x)),k = deg(h (x)), k = deg(h (x)).
2 2 1 1 2 2
LetC =hh(x)h (x)ibeaconstacycliccodewithshiftconstanta andparameters
1 1 1
[n ,m−k,d ] and letC =hh(x)h (x)i be a constacyclic code with shift constant
1 1 2 2
a and parameters [n ,m−k,d ]. Then an MT code C with a generator of the
2 2 2
form hp (x)h(x)h (x),p (x)h(x)h (x)i, where gcd(p(x),g(x)) = 1 for i = 1,2,
1 1 2 2 i
hasparameters[n +n ,m−k,d],where d ≥d +d .
1 2 1 2
PROOF. The assertion on dimension can be proven as in the previous theorems.
Toseewhyd ≥d +d ,letc:= f(x)·(p (x)h(x)h (x),p (x)h(x)h (x)),forsome
1 2 1 1 2 2
f(x)∈F [x], be an arbitrary codeword inC. We observe that the first component
q
of c is 0 if and only if g(x)|f(x) if and only if the second component of c is 0.
Therefore, in C there are no codewords with only 1 non-zero component, and
henced ≥d +d .
1 2
We now give an example of an MT code with parameters of a best-known
code, obtained using the theorem above. It shows that the actual minimum dis-
tanceofan MTcodecan besignificantlylargerthan thetheoretical lowerbound.
Example5.4. Let q=3,n =20,n =40,a =2and a =1 inTheorem 5.3. We
1 2 1 2
havegcd(xn1−a ,xn2−a )=x20−2=g(x)h(x)whereg(x)=x6+x5+x4+2x+
1 2
2. The constacyclic codes C and C generated by hh(x)h (x)i and hh(x)h (x)i
1 2 1 2
8
as described in Theorem 5.3 have parameters [20,6,9] and [40,6,18] respecti-
vely. Then the MT codeC with a generator in the form given by the theorem has
parameters [60,6,d] with d ≥ 27. We found that for p (x) = x3+x2+2x and
1
p (x)=x5+x4+x3+x2+2x+1, the minimum weight is actually36. Hence we
2
obtainaternary[60,6,36]codewhichisabestknowncodeforitsparameters[1].
Another way to obtain codes with larger minimum distances is to consider
subcodesofMTcodes.
Theorem 5.5. LetC =hg (x)ibeaconstacycliccodewithshiftconstanta and
1 1 1
parameters [n ,k,d ] and letC =hg (x)i be a constacyclic code with shift con-
1 1 2 2
stant a and parameters [n ,k,d ]. Let xn1 −a = g (x)h (x) and xn2 −a =
2 2 2 1 1 1 2
g (x)h (x)withdeg(h (x))=deg(h (x))=k. Then anMTcodeC withagenera-
2 2 1 2
tor of the form hp (x)g (x),p (x)g (x)i, where gcd(p (x),h (x))=1 for i=1,2,
1 1 2 2 i i
hasa subcodeC′ with parameters[n,k,d],where n=n +n andd ≥d +d .
1 2 1 2
PROOF. WeconsiderthesubcodeC′ ofC ofdimensionk, generated by
{xi·(p (x)g (x),p (x)g (x)):0≤i≤k−1},
1 1 2 2
where xi·(p (x)g (x),p (x)g (x)) = (xip (x)g (x),xip (x)g (x)). We have that
1 1 2 2 1 1 2 2
everycodewordofC′ is oftheform
a(x)·(p (x)g (x),p (x)g (x))=(a(x)p (x)g (x),a(x)p (x)g (x)),
1 1 2 2 1 1 2 2
where a(x) is a polynomial of degree < k. To prove d ≥ d +d , it suffices
1 2
to show that for a non-zero polynomial a(x) both components of the codeword
(a(x)p (x)g (x),a(x)p (x)g (x)) arenon-zero. But thisistruesincedeg(a(x))<
1 1 2 2
k = deg(h ) and gcd(p (x),h (x)) = 1, and therefore xn1 −a does not divide
1 1 1 1
a(x)p (x)g (x). Similarly,xn2−a doesnot dividea(x)p (x)g (x).
1 1 2 2 2
Note that a straightforward generalization of the argument above proves the
following
Theorem 5.6. Let C =hg(x)i be a constacyclic code with shift constant a and
i i i
parameters[n,k,d]fori=1,...,ℓ. Letxni−a =g (x)h(x)with deg(h(x))=k.
i i i i i i
Then anMT codeC with a generatoroftheform
hp (x)g (x),p (x)g (x),...,p (x)g (x)i,
1 1 2 2 ℓ ℓ
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wheregcd(p (x),h(x))=1fori=1,...,ℓ,hasasubcodewithparameters[n,k,d],
i i
i=ℓ i=ℓ
(cid:229) (cid:229)
wheren= n andd ≥ d .
i i
i=1 i=1
Finally,wehighlightaspecialcase ofTheorem5.5 whichmaybeuseful.
Corollary5.7. Let n <n , and xn2−a =g(x)h(x), where deg(h(x))=n . Let
1 2 2 1
p (x) and p (x) be any two polynomials such that gcd(p (x),xn1−a ) = 1 and
1 2 1 1
gcd(p (x),h(x))=1. LetCbetheMTsubcodegeneratedby{xi·(p (x),g(x)p (x)):
2 1 2
0 ≤ i < n }. Then C has parameters [n,k,d], where n = n +n ,k = n , and
1 1 2 1
d ≥d +1,whered istheminimumweightoftheconstacycliccodeoflengthn ,
2 2 2
shiftconstanta , andgeneratedbyg(x)p (x).
2 2
6. Examples
Finally, we present a few examples of subcodes of MT codes with good pa-
rameters. All of these codes have parameters of a best-known code, and some of
them are optimal. Moreover, some of them cannot be obtained as a constacyclic
orQT code. Consideringtheclass ofMT codesand theirsubcodesin acomputer
searchtodiscovernewlinearcodesisbothmorepromisingandmorechallenging
thanin theQT casebecause thesearch spaceislarger.
Example6.1. Let q = 7, n = 7, n = 16, and a = a = 1. We have gcd(x7−
1 2 1 2
1,x16−1)=1. Bylettingg(x)=x9+6x8+x7+2x6+x5+5x4+3x3+x2+2x+6,
p (x) = x6+2x5+x4+2x3+4x2, and p (x) = 2x6+6x4+x3+x2+6x+1 in
1 2
Corollary 5.7, we obtain a code with parameters [23,7,13] . According to the
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database[1], thisis theparametersofa best-knowncode.
Example6.2. Let q = 7, n = 4, n = 50, a = 2 and a = 3. We have d(x) =
1 2 1 2
gcd(x4−2,x50−3)=x2−3. Inthiscase,thegcdisgreaterthan1butwecanstill
take g (x)=1 as the generator of a constacyclic code so thatC =h1i is the tri-
1 1
vialcode[4,4,1]. Letg (x)=x46+x45+5x44+5x43+5x42+3x41+3x39+2x38+
2
x37+4x36+x35+2x34+2x33+5x32+3x31+4x30+6x29+2x27+5x26+x25+
6x24+3x22+5x21+5x20+6x19+x17+2x16+x15+5x14+6x13+4x12+6x11+
2x10+5x9+2x8+2x7+4x5+6x4+4x3+5x2+3x+2. Then g (x)|(x50−3)
2
and it generates a constacyclic codeC with parameters [50,4,42]. The MT sub-
2
codegeneratedby{xi·(g (x),p (x)g (x)):0≤i≤3},where p (x)=x3+2x2+
1 2 2 2
x+5, which is relatively prime with the check polynomial ofC , has parameters
2
[54,4,44] . Thisturnsoutto beanoptimalcode[1].
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