A Generalization of Quasi-twisted Codes: Multi-twisted codes
aa r X i v : . [ c s . I T ] J a n A Generalization of Quasi-twisted Codes:Multi-twisted codes
Nuh Aydin a , Ajdin Halilovi´c b a Kenyon College, Department of Mathematics, Gambier, OH 43022 b Lumina-The University of South-East Europe, S , os. Colentina 64b, 021187 Bucharest, Romania Abstract
Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes,have a special place in algebraic coding theory. Among other things, many of thebest-known or optimal codes have been obtained from these classes. In this workwe introduce a new generalization of QT codes that we call multi-twisted (MT)codes and study some of their basic properties. Presenting several methods ofconstructing codes in this class and obtaining bounds on the minimum distances,we show that there exist codes with good parameters in this class that cannot beobtained as QT or constacyclic codes. This suggests that considering this largerclass in computer searches is promising for constructing codes with better para-meters than currently best-known linear codes. Working with this new class ofcodes motivated us to consider a problem about binomials over finite fields and todiscover a result that is interesting in its own right.
Keywords: constacyclic codes, quasi-twisted codes, best-known codes
1. Introduction and Motivation
Every linear code over a finite field F q has three basic parameters: the length( n ), the dimension ( k ), and the minimum distance ( d ) that determine the quality ofthe code. One of the most important and challenging problems of coding theory isa discrete optimization problem: determine the optimal values of these parametersand construct codes whose parameters attain the optimal values. This optimizationproblem is very difficult. In general, it is only solved for the cases where either k or n − k is small. There is a database of best known linear codes with upper boundson minimum distances that is available online [1]. The database is updated as newcodes are discovered and reported by researchers. Preprint submitted to Finite Fields and Their Applications 5 ianuarie 2017 omputers are often used in searching for codes with best parameters but thereis an inherent difficulty: computing the minimum distance of a linear code iscomputationally intractable (NP-hard) [11]. Since it is not possible to conductexhaustive searches for linear codes if the dimension is large, researchers oftenfocus on promising subclasses of linear codes with rich mathematical structures.A promising and fruitful approach has been to focus on the class of quasi-twisted(QT) codes which includes cyclic, constacyclic, and quasi-cyclic (QC) codes asspecial cases. This class of codes is known to contain many codes with goodparameters. In the last few decades, a large number of record-breaking QC andQT codes have been constructed (e.g. [2]-[10]). The search algorithm introducedin [4] has been highly effective and used in several subsequent works ([5]-[10]).In this work, we introduce a new generalization of QT codes that we call multi-twisted (MT) codes. It turns out that this class also generalizes more recentlyintroduced classes of double cyclic codes ([2, 3]), QCT codes ([15]), and GQCcodes ([14]). After deriving some of their algebraic properties and obtaining alower bound on the minimum distance, we show that from this class we can obtainlinear codes with best-known or optimal parameters that cannot be obtained fromthe smaller classes of constacyclic or QT codes.Before introducing this new class of codes, we recall some fundamental resultsabout constacyclic and QT codes that will be needed later.
2. Constacyclic and Quasi-twisted Codes
Constacyclic codes are very well-known in algebraic coding theory. Let a ∈ F ∗ q = F q \ { } . A linear code C over a finite field F q is called constacyclic with shiftconstant a if it is closed under the constacyclic shift, i.e. for any ( c , c , . . . , c n − ) ∈ C , T a ( c , c , . . . , c n − ) : = ( c n − , c , c , . . . , c n − ) ∈ C . When a =
1, we obtain thevery important special case of cyclic codes. Many well-known codes are instancesof cyclic codes.Under the usual isomorphism p : F nq → F q [ x ] / h x n − a i , where p ( u ) = u + u x + u x + . . . + u n − x n − , for u = ( u , u , . . . , u n − ) ∈ F nq , it is well-known thata constacyclic code is an ideal in the ring F q [ x ] / h x n − a i . Moreover, for everyconstacyclic code C there is a unique, monic polynomial of least degree in C that generates C , i.e. C = h g ( x ) i = { f ( x ) g ( x ) mod x n − a : f ( x ) ∈ F q [ x ] } . Thisstandard generator is a divisor of x n − a , so that x n − a = g ( x ) h ( x ) , for some h ( x ) ∈ F q [ x ] which is called the check polynomial of C . Note that the set of all codewordscan be described as C = { f ( x ) g ( x ) : f ( x ) ∈ F q [ x ] and deg ( f ( x )) < deg ( h ( x )) } , i.e.the set { g ( x ) , xg ( x ) , x g ( x ) , . . . , x k − g ( x ) } is a basis for C where k = deg ( h ( x )) .2 constacyclic code has many other generators as well and we have a completecharacterization of them as follows. Lemma 2.1. [4] Let C = h g ( x ) i be a constacyclic code of length n and shift con-stant a with canonical generator g ( x ) and check polynomial h ( x ) so that x n − a = g ( x ) h ( x ) . Then C = h g ′ ( x ) i if and only if g ′ ( x ) is of the form g ( x ) p ( x ) with gcd ( p ( x ) , h ( x )) = . A linear code C is said to be ℓ -quasi-twisted ( ℓ -QT) if, for a positive in-teger ℓ , it is invariant under T ℓ a , that is, whenever ( c , c , . . . , c n − ) ∈ C , then ( ac n − ℓ , . . . , ac n − , c , c , . . . , c n − ℓ − ) ∈ C as well. It is important to note that whengcd ( ℓ, n ) = ℓ | n , thecase ℓ = C of length n = m ℓ with shiftconstant a is an R -submodule of R ℓ where R = F q [ x ] / h x m − a i . If C has a singlegenerator of the form ( f ( x ) , f ( x ) , . . . , f ℓ ( x )) then it is called a 1-generator QTcode, otherwise a multi-generator QT code. Most of the literature on QT codesfocuses on the 1-generator case. We will do the same in this work.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 followingtheorem. Theorem 2.2 (see [4]).
Let C be a -generator ℓ -QT code of length n = m ℓ witha generator of the form: ( f ( x ) g ( x ) , f ( x ) g ( x ) , · · · , f ℓ ( x ) g ( x )) , (1) where g ( x ) , f i ( x ) ∈ F q [ x ] / h x m − a i , such that x m − a = g ( x ) h ( x ) and f i ( x ) is re-latively prime to h ( x ) for each i = , . . . , ℓ . Then C is an [ n , k , d ′ ] q -code wherek = m − deg ( g ( x )) , d ′ ≥ ℓ · d ( C g ) , and d ( C g ) denotes the minimum distance of theconstacyclic code of length m generated by g ( x ) . A proof of this theorem is given in [4]. In reality, the actual minimum dis-tance of C is often considerably larger than the lower bound given by the theorem.Researchers designed algorithms and conducted computer searches based on thistheorem and discovered many new linear codes ([5]-[10]).3 . Multi-twisted Codes We propose an even more general class of linear codes, which we call multi-twisted (MT) codes. A multi-twisted module V is an F q [ x ] -module of the form V = ℓ (cid:213) i = F q [ x ] / h x m i − a i i , where a i ∈ F q \ { } and m i are (possibly distinct) positive integers. An MT codeis an F q [ x ] -submodule of a multi-twisted module V . Equivalently, we can definean MT code in terms of the shift of a codeword. Namely, a linear code C is multi-twisted if for any codeword ~ c = ( c , , . . . , c , m − ; c , , . . . , c , m − ; . . . ; c ℓ, , . . . , c ℓ, m ℓ − ) ∈ C , its multi-twisted shift ( a c , m − , c , , . . . , c , m − ; a c , m − , c , , . . . , c , m − ; . . . ; a ℓ c ℓ, m ℓ − , . . . , c ℓ, m ℓ − ) is also a codeword. If we identify a vector ~ c with C ( x ) = ( c ( x ) , c ( x ) , . . . , c ℓ ( x )) where c i ( x ) = c i , + c i , x + · · · + c i , m i − x m i − , then the MT shift corresponds to xC ( x ) = ( xc ( x ) mod x m − a , . . . , xc ℓ ( x ) mod x m ℓ − a ℓ ) .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 aspecial case when m = m = · · · = m ℓ and a = a = · · · = a ℓ . Moreover, morerecently introduced classes of codes called generalized quasi-cyclic (GQC) codes[14], QCT codes [15], and double cyclic codes [12, 13] can be viewed as specialcases of MT codes.An MT code is a one-generator code if it is generated by a single element of V . This work will focus primarily on one-generator MT codes.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 toconsidering the greatest common divisor of two binomials x n − a and x n − a .Working on this question, we discovered a result about the greatest common di-visor of two binomials x n − a and x n − a over F q which we believe is a newresult about polynomials over finite fields, and interesting in its own right. Westate and prove this result in the next section.4 . A Result About Binomials over Finite Fields Considering generators of MT codes brings up the problem of determining thegreatest common divisor of two binomials of the form x n − a and x n − a . Wefound that the gcd of two such polynomials is either 1, or another binomial of thesame form. The precise statement and a proof are as follows. Theorem 4.1.
Let P = { f ∈ F q [ x ] : f = x m − a with a ∈ F ∗ q , m ∈ N , gcd ( m , q ) = } . Then for f , g ∈ P, gcd ( f , g ) is either 1 or of the form x gcd ( deg ( f ) , deg ( g )) − a, for somea ∈ F q . In particular, gcd ( f , g ) ∈ P ∪ { } . P ROOF . We will let ord m ( q ) denote the multiplicative order of q mod m , i.e. itis the smallest positive integer k such that q k ≡ m . For a non-zero element a ∈ F ∗ q , | a | denotes the order of a in the multiplicative group of ( F ∗ q , · ) . Let f = x n − a and g = x n − a for some n , n ∈ N and non-zero elements a , a ∈ F q with r = | a | and r = | a | . Let s = ord n r ( q ) and s = ord n r ( q ) . It is knownthat f and g split into linear factors in the extensions F q s and F q s , respectively. Inorder to find a common extension of these two fields, consider s = ord n r n r ( q ) .We claim that F q s , F q s ⊆ F q s , that is, s | s and s | s . Indeed, since n r n r | q s − n r | q s −
1. Since s = ord n r ( q ) , it follows that s | s .Similarly we obtain s | s .Let z ∈ F q s be a primitive n n th root of unity. Then z n and z n are primitive n th and n th roots of unity, respectively.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 throot of a and n th root of a . It follows that the roots of f are d , dz n , d ( z n ) , . . . , d ( z n ) n − and d ( z n ) n − , and the roots of g are d , dz n , d ( z n ) , . . . , d ( z n ) n − and d ( z n ) n − . The set of roots of gcd ( f , g ) is the intersection of the two sets above. Note thatthese sets are actually cosets of the multiplicative subgroups { , z n , ( z n ) , . . . , ( z n ) n − } and { , z n , ( z n ) , . . . , ( z n ) n − } of F q s , respecti-vely. It follows that the set of roots of gcd ( f , g ) is a coset of the intersection ofthese subgroups. Therefore, z n n / 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 ) d , dz n n / d , d ( z n n / d ) , . . . , d ( z n n / d ) d − . Therefore, deg ( gcd ( f , g )) = gcd ( n , n ) = gcd ( deg ( f ) , deg ( g )) . Finally, we showthat a : = d d ∈ F q to prove that gcd ( f , g ) is of the desired form x m − a , hencecompleting the proof. Write n = dt and n = dt for some relatively primeintegers t , t . Since gcd ( t , t ) =
1, there exist integers u , v such that ut + vt = d dt = a and d dt = a , which implies a = d d = d udt + vdt = a u · a v ∈ F q . Example 4.2.
Over F , gcd ( x − , x − ) = x − , gcd ( x − , x − ) = x − and gcd ( x − , x − ) = . This theorem has some implications for constacyclic codes. Suppose a , a ∈ F q and n , n ∈ Z + are such that gcd ( x n − a , x n − a ) = x m − a for some non-zero m ∈ Z and a ∈ F q . Any constacyclic code C of length m with shift constant a has a generator g ( x ) that divides x m − a , hence g ( x ) | ( x n − a ) and g ( x ) | ( x n − a ) . Therefore, we observe that the polynomial g ( x ) can also be regarded as the(standard) generator of a constacyclic code C of length n with shift constant a as well as the generator of a constacyclic code C of length n with shift constant a .
5. More on Constructions of MT Codes and Their Parameters
We frequently have gcd ( x n − a , x n − a ) =
1. Consider a 1-generator MTcode C in this case with a generator of the form h g ( x ) , g ( x ) i where g ( x ) | x n − a and g ( x ) | x n − a . Clearly, gcd ( g ( x ) , g ( x )) = C = h g ( x ) i with parameters [ n , k , d ] and C = h g ( x ) i with parameters [ n , k , d ] , then weobserve that C has parameters [ n + n , k + k , min { d , d } ] . We formally provethis in the next theorem. Theorem 5.1.
Let n , n ∈ Z + , a , a ∈ F ∗ q be such that gcd ( x n − a , x n − a ) = .Let x n − a = g ( x ) h ( x ) and x n − a = g ( x ) h ( x ) with k = deg ( h ( x )) , k = deg ( h ( x )) . Let C = h g ( x ) i and C = h g ( x ) i with parameters [ n , k , d ] and [ n , k , d ] , respectively. Then the MT code C with generator h g ( x ) , g ( x ) i hasparameters [ n , k , d ] , where n = n + n , k = k + k , and d = min { d , d } . P ROOF . The assertions on length of the MT code C is clear. To prove the asser-tion on dimension, we show that the set S = { x i · ( g ( x ) , g ( x )) : 0 ≤ i ≤ k − } ,6here k = k + k , is a basis for C . Suppose that f ( x ) · ( g ( x ) , g ( x )) = ( f ( x ) g ( x ) mod x n − a , f ( x ) g ( x ) mod x n − a ) = . Then f ( x ) g ( x ) ≡ x n − a and f ( x ) g ( x ) ≡ x n − a . Therefore, h ( x ) | f ( x ) and h ( x ) | f ( x ) . Since h ( x ) and h ( x ) are relatively prime (because gcd ( x n − a , x n − a ) = h ( x ) h ( x ) | f ( x ) , and hence deg ( f ( x )) ≥ k . This implies the linear independenceof vectors in S .It remains to show that S is a set of generators for C . For this it sufficesto show that for every f ( x ) ∈ F q [ x ] with deg ( f ( x )) ≥ k there exists r ( x ) ∈ F q [ x ] with deg ( r ( x )) < k such that f ( x ) · ( g ( x ) , g ( x )) = r ( x ) · ( g ( x ) , g ( x )) . So let f ( x ) ∈ F q [ x ] be an arbitrary polynomial with deg ( f ( x )) ≥ k . We can write f ( x ) as f ( x ) = h ( x ) h ( x ) q ( x ) + r ( x ) , for some q ( x ) , r ( x ) ∈ F q [ x ] where deg ( r ( x )) < deg ( h ( x ) h ( x )) = k . It follows that f ( x ) · ( g ( x ) , g ( x )) = ( f ( x ) g ( x ) mod x n − a , f ( x ) g ( x ) mod x n − a )= ( r ( x ) g ( x ) mod x n − a , r ( x ) g ( x ) mod x n − a )= r ( x ) · ( g ( x ) , g ( x )) . Finally, to show that d = min { d , d } , let u ( x ) = t ( x ) g ( x ) ∈ C be a codewordof minimum weight in C . Since gcd ( h ( x ) , h ( x )) =
1, we have C = h g ( x ) i = h h ( x ) g ( x ) i . Hence, u ( x ) = t ′ ( x ) h ( x ) g ( x ) for some t ′ ( x ) ∈ F q [ x ] (degree ofwhich can be taken to be < k ). Letting f ( x ) = t ′ ( x ) h ( x ) and considering thecodeword f ( x ) · ( g ( x ) , g ( x )) , we find a codeword ( u ( x ) , ) ∈ C of weight d in C . We can similarly show that there exists a codeword of weight d in C as well.Given this result, we cannot hope to find codes with good parameters underthe conditions of the above proposition. We need to look for alternative ways toconstruct MT codes with potentially high minimum distances. Before coming tomore promising constructions, let us first disqualify another case where no MTcodes with high minimum distances can be expected. Theorem 5.2.
Let a , a ∈ F ∗ q , n = n ∈ Z + be such that gcd ( x n − a , x n − a ) = x m − a = g ( x ) h ( x ) , for some m ∈ Z + , a ∈ F q and let x n − a = g ( x ) h ( x ) h ( x ) andx n − a = g ( x ) h ( x ) h ( x ) with k = deg ( h ( x )) , k = deg ( h ( x )) , k = deg ( h ( x )) .Let C be the MT code of length n + n with generator h g ( x ) p ( x ) , g ( x ) p ( x ) i ,where gcd ( h ( x ) h ( x ) , p ( x )) = and gcd ( h ( x ) h ( x ) , p ( x )) = . Then C has pa-rameters [ n + n , k + k + k , d ] with d ≤ . P ROOF . An argument identical to the one used in Proposition 5.1 shows that S = x i · ( g ( x ) p ( x ) , g ( x ) p ( x )) : 0 ≤ i ≤ k + k + k − } is a basis for C , and henceproves the assertion about the dimension.In order to see that minimum distance is at most 2, we first assume, w.l.o.g.,that n > n and observe that C is also generated by h g ( x ) , g ( x ) p ( x ) i . Then wenote that h ( x ) h ( x ) · ( g ( x ) , g ( x ) p ( x )) = ( x n − a , ) , which is a codeword of wei-ght 2.Note that the theorem above points out a significant difference between QTand MT codes, since many good codes from the class of QT codes have genera-tors of the form h g ( x ) , g ( x ) p ( x ) i , where gcd ( h ( x ) , p ( x )) = Theorem 5.3.
Let a , a ∈ F ∗ q , n , n ∈ Z + be such that gcd ( x n − a , x n − a ) = x m − a = g ( x ) h ( x ) , for some m ∈ Z + , a ∈ F q and let x n − a = g ( x ) h ( x ) h ( x ) andx n − a = g ( x ) h ( x ) h ( x ) with k = deg ( h ( x )) , k = deg ( h ( x )) , k = deg ( h ( x )) .Let C = h h ( x ) h ( x ) i be a constacyclic code with shift constant a and parameters [ n , m − k , d ] and let C = h h ( x ) h ( x ) i be a constacyclic code with shift constanta and parameters [ n , m − k , d ] . Then an MT code C with a generator of theform h p ( x ) h ( x ) h ( x ) , p ( x ) h ( x ) h ( x ) i , where gcd ( p i ( x ) , g ( x )) = for i = , ,has parameters [ n + n , m − k , d ] , where d ≥ d + d . P ROOF . The assertion on dimension can be proven as in the previous theorems.To see why d ≥ d + d , let c : = f ( x ) · ( p ( x ) h ( x ) h ( x ) , p ( x ) h ( x ) h ( x )) , for some f ( x ) ∈ F q [ x ] , be an arbitrary codeword in C . We observe that the first componentof 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, andhence d ≥ d + d .We now give an example of an MT code with parameters of a best-knowncode, obtained using the theorem above. It shows that the actual minimum dis-tance of an MT code can be significantly larger than the theoretical lower bound. Example 5.4.
Let q = , n = , n = , a = and a = in Theorem 5.3. Wehave gcd ( x n − a , x n − a ) = x − = g ( x ) h ( x ) where g ( x ) = x + x + x + x + . The constacyclic codes C and C generated by h h ( x ) h ( x ) i and h h ( x ) h ( x ) i s described in Theorem 5.3 have parameters [ , , ] and [ , , ] respecti-vely. Then the MT code C with a generator in the form given by the theorem hasparameters [ , , d ] with d ≥ . We found that for p ( x ) = x + x + x andp ( x ) = x + x + x + x + x + , the minimum weight is actually 36. Hence weobtain a ternary [ , , ] code which is a best known code for its parameters [1]. Another way to obtain codes with larger minimum distances is to considersubcodes of MT codes.
Theorem 5.5.
Let C = h g ( x ) i be a constacyclic code with shift constant a andparameters [ n , k , d ] and let C = h g ( x ) i be a constacyclic code with shift con-stant a and parameters [ n , k , d ] . Let x n − a = g ( x ) h ( x ) and x n − a = g ( x ) h ( x ) with deg ( h ( x )) = deg ( h ( x )) = k. Then an MT code C with a genera-tor of the form h p ( x ) g ( x ) , p ( x ) g ( x ) i , where gcd ( p i ( x ) , h i ( x )) = for i = , ,has a subcode C ′ with parameters [ n , k , d ] , where n = n + n and d ≥ d + d . P ROOF . We consider the subcode C ′ of C of dimension k , generated by { x i · ( p ( x ) g ( x ) , p ( x ) g ( x )) : 0 ≤ i ≤ k − } ,where x i · ( p ( x ) g ( x ) , p ( x ) g ( x )) = ( x i p ( x ) g ( x ) , x i p ( x ) g ( x )) . We have thatevery codeword of C ′ is of the form a ( x ) · ( p ( x ) g ( x ) , p ( x ) g ( x )) = ( a ( x ) p ( x ) g ( x ) , a ( x ) p ( x ) g ( x )) , where a ( x ) is a polynomial of degree < k . To prove d ≥ d + d , it sufficesto 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 )) are non-zero. But this is true since deg ( a ( x )) < k = deg ( h ) and gcd ( p ( x ) , h ( x )) =
1, and therefore x n − a does not divide a ( x ) p ( x ) g ( x ) . Similarly, x n − a does not divide a ( x ) p ( x ) g ( x ) .Note that a straightforward generalization of the argument above proves thefollowing Theorem 5.6.
Let C i = h g i ( x ) i be a constacyclic code with shift constant a i andparameters [ n i , k , d i ] for i = , . . . , ℓ . Let x n i − a i = g i ( x ) h i ( x ) with deg ( h i ( x )) = k.Then an MT code C with a generator of the form h p ( x ) g ( x ) , p ( x ) g ( x ) , . . . , p ℓ ( x ) g ℓ ( x ) i , here gcd ( p i ( x ) , h i ( x )) = for i = , . . . , ℓ , has a subcode with parameters [ n , k , d ] ,where n = i = ℓ (cid:229) i = n i and d ≥ i = ℓ (cid:229) i = d i . Finally, we highlight a special case of Theorem 5.5 which may be useful.
Corollary 5.7.
Let n < n , and x n − a = g ( x ) h ( x ) , where deg ( h ( x )) = n . Letp ( x ) and p ( x ) be any two polynomials such that gcd ( p ( x ) , x n − a ) = and gcd ( p ( x ) , h ( x )) = . Let C be the MT subcode generated by { x i · ( p ( x ) , g ( x ) p ( x )) :0 ≤ i < n } . Then C has parameters [ n , k , d ] , where n = n + n , k = n , andd ≥ d + , where d is the minimum weight of the constacyclic code of length n ,shift constant a , and generated by g ( x ) p ( x ) .
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 ofthem are optimal. Moreover, some of them cannot be obtained as a constacyclicor QT code. Considering the class of MT codes and their subcodes in a computersearch to discover new linear codes is both more promising and more challengingthan in the QT case because the search space is larger.
Example 6.1.
Let q = , n = , n = , and a = a = . We have gcd ( x − , x − ) = . By letting g ( x ) = x + x + x + x + x + x + x + x + x + ,p ( x ) = x + x + x + x + x , and p ( x ) = x + x + x + x + x + inCorollary 5.7, we obtain a code with parameters [ , , ] . According to thedatabase [1], this is the parameters of a best-known code. Example 6.2.
Let q = , n = , n = , a = and a = . We have d ( x ) = gcd ( x − , x − ) = x − . In this case, the gcd is greater than but we can stilltake g ( x ) = as the generator of a constacyclic code so that C = h i is the tri-vial code [ , , ] . Let g ( x ) = x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + . Then g ( x ) | ( x − ) and it generates a constacyclic code C with parameters [ , , ] . The MT sub-code generated by { x i · ( g ( x ) , p ( x ) g ( x )) : 0 ≤ i ≤ } , where p ( x ) = x + x + x + , which is relatively prime with the check polynomial of C , has parameters [ , , ] . This turns out to be an optimal code [1]. xample 6.3. Let q = , n = , n = , a = and a = . The cyclic codeC of length 19 generated by g ( x ) = x + has parameters [ , , ] . The con-stacyclic code of length 34 and shift constant 2 generated by g ( x ) = x + x + x + x + x + x + x + x + x + x + x + x + x + has para-meters [ , , ] . By Theorem 5.5 we know that an MT subcode generated by { x i · ( g ( x ) , p ( x ) g ( x )) : 0 ≤ i ≤ } , where gcd ( p ( x ) , h ( x )) = will have para-maters [ , , d ] with d ≥ . We have found that for p ( x ) = x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + the resulting code C has parameters [ , , ] , which is the parameters of a bestknown code [1]. We would like to point out that since n = is a prime number,it is not possible to obtain a code of length 53 and dimension 18 from the classof QT codes with index greater than 1. Moreover, from the factorization of thepolynomial x − a, for any a ∈ F ∗ , we see that it is not possible to obtain a codewith these parameters from the class of constacyclic codes either. Example 6.4.
Let q = , n = , n = , a = , and a = . The 1-generatorternary MT subcode C of dimension 12 generated by { x i · ( g ( x ) , p ( x ) g ( x )) :0 ≤ i ≤ } , where g ( x ) = x + , g ( x ) = x + x + x + x + x + x + x + x + , and p ( x ) = x + x + x + x + x + x + x + x + x + x + x, hasparameters [ , , ] , which means that C is a best-known code for its parameterset [1]. From the factorizations of x − and x − we observe that neither acyclic nor a constacyclic code exists for length 33 and dimension 12. Moreover,since = · , a 1-generator QT code with these parameters does not existeither. ReferencesBibliografie GF ( ) from quasi-twisted codes, toappear in Adv. Math. Commun. (2016).[9] N. Aydin, N. Connolly, J. Murphree, New binary linear codes from quasi-cyclic codes and an augmentation algorithm, preprint.[10] R. Ackerman, and N. Aydin, New quinary linear codes from quasi-twistedcodes and their duals, Appl. Math. Letters 24(4) (2011) 512-515.[11] A. Vardy, The Intractability of computing the minimum distance of a code,IEEE Trans. Inf. Theory 43 (1997) 1757-1766.[12] J. Borges, C. Fernandez-Cordoba, R. Ten-Valls, Z -double cyclic codes, pre-print, arXiv: 1410.5604v1 (2014).[13] J. Gao, M. Shi, T. Wu, F. Fu, On double cyclic codes over Z4