New Extremal binary self-dual codes of length 68 from a novel approach to neighbors
aa r X i v : . [ m a t h . C O ] F e b NEW EXTREMAL BINARY SELF-DUAL CODES OF LENGTH 68 FROM A NOVELAPPROACH TO NEIGHBORS
JOE GILDEA, ABIDIN KAYA, ADRIAN KORBAN AND BAHATTIN YILDIZ
Abstract.
In this work, we introduce the concept of distance between self-dual codes, which generalizes theconcept of a neighbor for self-dual codes. Using the k -neighbors, we are able to construct extremal binaryself-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codesof length 68 with new weight enumerators including 42 codes with γ = 8 in their W , and 40 with γ = 9 intheir W , . These examples are the first in the literature for these γ values. This completes the theoreticallist of possible values for γ in W , . Introduction
Self-dual codes are a special class of linear codes. Because of the many interesting properties that theyhave and the many different fields that they are connected with, they have attracted a considerable interestin coding theory research community.One of the most active research areas in the field of self-dual codes is the construction and classificationof extremal binary self-dual codes. Type I extremal binary self-dual codes of lengths such as 64, 66, 68, etc.have parameters in their weight enumerators, which have not all been found to exist. Hence, the rcent yearshave seen a surge of activity in finding extremal binary self-dual codes of various lengths with new weightenumerators. Many different techniques have been employed in constructing these extremal binary self-dualcodes such as constructions over certain rings, constructions through automorphism groups, neighboringconstructions, shadows, extensions, etc. [1], [4], [7], [8], [10], [13], [14] are just a sample of the works thatcontain these ideas and their applications in finding new extremal binary self-dual codes.In this work, we introduce the concept of “distance” between self-dual codes. After proving some theo-retical results about the distance we observe that the neighbor can be defined in terms of the distance andthis leads to the concept of “ k -range neighbors” or “ k -neighbors”, which generalize the concept of neighborsof self-dual codes. We then use these k -neighbors to construct extremal self-dual codes of length 68 from agiven self-dual code. In particular we construct 139 new extremal binary self-dual codes of length 68 withnew weight enumerators, including the first examples with γ = 8 , W , in the literature. This completesthe list of possible γ values that can be found in W , . 42 of the codes we have constructed have γ = 8 intheir weight enumerator, while 40 of them have γ = 9 in their weight enumerators.The rest of the work is organized as follows: In section 2, we give the preliminaries about self-dualcodes and the neighbor construction.In section 3, we introduce the concept of distance and define the relatedgeneralization of the neighbors. In section 4, we apply the generalized neighbors to construct extremal binaryself-dual codes of length 68 with new weight enumerators. We finish the work with concluding remarks anddirections for possible future research. 2. Preliminaries
Self-dual codes.
For x = ( x , x , . . . , x n ) and y = ( y , y , . . . , y n ) ∈ F n , we define h x, y i = x y + x y + · · · + x n y n . This inner product leads to
Mathematics Subject Classification.
Primary 94B05; Secondary 11T71.
Key words and phrases. extremal self-dual codes, neighbor, distance of self-dual codes, k th neighbor, weight enumerator. Definition 2.1.
Let C be a binary linear code over of length n , then we define the dual of C as C ⊥ := { y ∈ F n |h y, x i = 0 , ∀ x ∈ C } . Note that, if C is a linear [ n, k ]-code, then C ⊥ is a linear [ n, n − k ]-code. Definition 2.2. If C ⊆ C ⊥ , then C is called self-orthogonal and it is called self-dual if C = C ⊥ . Definition 2.3.
Let C be a self-dual binary code. If the Hamming weights of all the codewords in C aredivisible by 4, C is called Type II (or doubly-even), otherwise it is called
Type I (or singly even).The following theorem gives an upper bound for minimum distance of self-dual codes:
Theorem 2.4. ([15] , [5]) Let d I ( n ) and d II ( n ) be the minimum distance of a Type I and Type II binarycode of length n . then d II ( n ) ≤ ⌊ n ⌋ + 4 and d I ( n ) ≤ (cid:26) ⌊ n ⌋ + 4 if n
22 (mod 24)4 ⌊ n ⌋ + 6 if n ≡
22 (mod 24) . Self-dual codes that attain the bounds given in the previous theorem are called extremal .2.2.
The neighbor construction.
Two self-dual codes of length n are called neighbors if their intersectionis a code of dimension n −
1. This idea has been used extensively in the literature to construct new self-dualcodes from an existing one. For some of the works that have used this idea, we can refer to [3], [7], [10] andreferences therein.Given a self-dual code C , a vector x ∈ F n − C is picked and then D is formed by letting D = D h x i ⊥ ∩ C, x E .The search for D can be made efficient by using the standard form of the generator matrix of C , which letsone to fix the first n/ x without loss of generality. Usually in practical applications the first n/ x are set to be 0.3. Distance between self-dual codes and generalized neighbors
To generalize the notion of a neighbor, we first begin with the following definition of a distance betweentwo self-dual codes:
Definition 3.1.
Let C and C be two binary self-dual codes of length n . The neighbor-distance between C and C is defined as d N ( C , C ) = n − dim ( C ∩ C ) . Proposition 3.2. d N is a metric on the set of all binary self-dual codes of length n .Proof. Since dim ( C ∩ C ) ≤ dim ( C ) = dim ( C ) = n , we have d N ( C , C ) ≥ C , C .Next, observe that if d N ( C , C ) = 0, this means dim ( C ∩ C ) = n dim ( C ) = dim ( C ) , which implies C = C = C ∩ C . Conversely, if C = C , then d N ( C , C ) = n − n = 0 . By the definition, it is clear that d N ( C , C ) = d N ( C , C ).For the triangle inequality, assume that C , C , C are self-dual codes. Observe that( C ∩ C ) ∪ ( C ∩ C ) = C ∩ ( C ∪ C ) ⊆ C . which implies dim ( C ∩ C ) + dim ( C ∩ C ) ≤ dim ( C ) = n . Thus we have dim ( C ∩ C ) + dim ( C ∩ C ) − dim ( C ∩ C ) ≤ n . ENERALIZED NEIGHBORS 3
Adding n to both sides and sending dim ( C ∩ C ) + dim ( C ∩ C ) over to the right side of the equation, weget d N ( C , C ) = n − dim ( C ∩ C ) ≤ n − dim ( C ∩ C ) + n − dim ( C ∩ C ) ≤ d N ( C , C ) + d N ( C , C ) . (cid:3) The next proposition shows that self-dual codes cannot have the maximum distance to each other:
Proposition 3.3.
Let C and C be two binary self-dual codes of length n . Then d N ( C , C ) < n .Proof. It is well known that if C is any binary self-dual code, then (1 , , . . . , ∈ C . thus 1 ∈ C ∩ C , whichimplies that dim ( C ∩ C ) ≥
1. But then d N ( C , C ) = n − dim ( C ∩ C ) ≤ n − < n . (cid:3) Question:
Is there an upper bound on the distance between two self-dual codes? The proposition showsthat the distance cannot be larger than n −
1. It is an open question whether this upper bound can bereduced further.We now define k -range neighbor and k -neighbor of a code: Definition 3.4.
Let C and C be two self-dual codes. C and C are said to be k -range neighbors if d N ( C , C ) ≤ k and they are called k -neighbors if d N ( C , C ) = k . Remark . The neighbor of a self-dual code is well known in the literature and it corresponds to a 1-neighborin our context.
Remark . The concept of a k -range neighbor code can be more useful than the strict k -neighbor codes,because of the following observation:Suppose C and C are self-dual binary codes with generator matrices [ I n/ | M ] and [ I n/ | M ], respec-tively, where M = r r r ... r n/ , M = s s s ... s n/ . Here r i and s j are the rows of M and M respectively. If r i = s i for i = k + 1 , k + 2 , . . . , n/
2, then C and C are k -range neighbors. Remark . As we observed above, the ordinary neighbor of a code C is a 1-neighbor. We can also observethat, the neighbor a 1-neighbor of C is a 2-range neighbor of C . This can be generalized into consideringthe neighbor of a neighbor of a neighbor etc. of a code as a k -range neighbor of the original code. JOE GILDEA, ABIDIN KAYA, ADRIAN KORBAN AND BAHATTIN YILDIZ Applications of k -range neighbor codes to extremal self-dual codes In this section we will give an equivalent description for the k -range neighbors and use them to constructnew extremal binary self-dual codes. Let N (0) be a binary self-dual code of length 2 n . Let x ∈ F n − N (0) ,define N ( i +1) = D h x i i ⊥ ∩ N ( i ) , x i E where N ( i +1) is the neighbour of N ( i ) and x i ∈ F n − N ( i ) .It is not hard to see that N ( i ) defined in this way is an i -range neighbor of N (0) as was observed abovein Remark 3.7. In what follows, we will apply this idea to search for extremal binary self-dual codes from k -range neighbors of a known code. We use Magma Algebra System ([2]) for our searches.4.1. Numerical results from i -range neighbours. The possible weight enumerator of an extremal binaryself-dual code of length 68 (of parameters [68 , , W , = 1 + (442 + 4 β ) y + (10864 − β ) y + · · · , ≤ β ≤ ,W , = 1 + (442 + 4 β ) y + (14960 − β − γ ) y + · · · where 0 ≤ γ ≤
9. Recently, Yankov et al. constructed the first examples of codes with a weight enumeratorfor γ = 7 in W , in [1]. Together with these, the existence of codes in W , is known for many values. Inorder to save space we only give the lists for γ = 5, γ = 6 and γ = 7, which are updated in this work; γ = 5 with β ∈ { ... ,182,187,189,191,192,193,201,202,213 } γ = 6 with β ∈ { , , , . . . , , , , , , } γ = 7 with β ∈ { m | m = 14 , . . . , , } Let N (0) be the extremal binary self-dual code of length 68 ( W , ) with the parameters γ = 5 and β = 213which was recently constructed in [9]. Its binary generating matrix is given by ( I | A ) where A = . Implementing the formula described above to this code N (0) , we obtain: ENERALIZED NEIGHBORS 5
Table 1. i -range neighbour of N (0) i N ( i +1) x i | Aut ( N ( i +1) ) | γ β N (1) (1100000101101111011001110100000100) 1 6 2101 N (2) (0111111111110110010100110111001100) 1 N (3) (0111010010010101001000101110011001) 1 N (4) (1000000111110011101001110001110000) 1 Neighbours of Neighbours.
In this section, we separately consider neighbours of N (0) , N (1) , N (2) , N (3) and N (4) . Table 2.
Neighbours of N (0) C i ( x , x , ..., x ) | Aut ( N ,i ) | γ β C (1001100000010100010100001111100011) 2 C (1000010001011010000011010000011010) 1 C (0111101000110110001011101100010000) 1 C (0111001101010010011001000101101010) 1 C (0100101101000111111110110101110111) 2 C (0011011100110001100010000000100100) 1 C (0111011111101001111101101111001000) 1 Table 3.
Neighbours of N (1) C i ( x , x , ..., x ) γ β C i ( x , x , ..., x ) γ β C (1001010111010111110011100111000011) C (0001011110111110011101001111111100) C (1011110110111010111010010111101111) C (1011011001101100010101001010001111) C (0111001000010101110001001100111100) C (0111111111011111101100100100001110) C (1011111001001110011110000010100011) C (1010100011111100010011111101001101) C (1011010111110011001011000100111011) C (0011110011110111111101101100110100) C (0000000010010101010011001010001011) C (1011101110011101111110100101011100) C (1111000000010110001111001111010101) C (1010111010101011100011100011001111) C (1000001101010100110101000011000101) C (0101011110100101000000011010001111) C (1111011101111101110110100010000111) C (1111011010010110011101100001000110) C (1011110001001000100001000110100000) C (1001010010110100100000001100000101) C (1010010010100011111100011100111010) C (0100010011101100010110001010110000) C (0100100000011110000010011000010110) C (0001110111010001111011010001011111) C (0010110010011001111110101000011110) C (0011001100100010111110000000011001) Table 4.
Neighbours of N (2) C i ( x , x , ..., x ) γ β C i ( x , x , ..., x ) γ β C (1101111111101111011110011101101110) C (1001101001000111010110110111111111) C (1011010010001010111100010010000100) C (1101111001010110110111010110001111) C (1011101100100101100110111101101111) C (1001111111011111110110010001001110) C (1101010001010110001001111001100010) C (0110111111000000011011000001110001) C (0010011000010000000011111111111010) C (1001101101101011111111010100101101) C (0110000001011010011001111110100010) C (1011000011111010100011111011100011) C (0100001110010111010110101010011110) C (0000001101111110100001100001000000) C (0100110110001011101001011000110001) C (1111010100000010111100100101110101) C (1100100000001001100110010111111111) C (0000110011100001111010110100110001) JOE GILDEA, ABIDIN KAYA, ADRIAN KORBAN AND BAHATTIN YILDIZ
Table 5.
Neighbours of N (3) C i ( x , x , ..., x ) γ β C i ( x , x , ..., x ) γ β C (0101000010100010000111100101011100) C (0000011101001011001010111100001110) C (0000011101101010010000110000101000) C (1011010010100001010000111011100110) C (1011110011111111111100111110100111) C (1101111111010111111111100010110111) C (0000110100100011011001001101111010) C (1101000111000011001010010001000000) C (0000001011100100110100101111000100) C (0111011011011011010010101101100011) C (0011011011110001000111111111011110) C (1010111011000111100110111001110111) C (0010010011101001110101011010111100) C (0001001001010000111101111001110111) C (0001101011101101100010111110110011) C (0100100001010000001010001111100010) C (1011100101000101100011110111101011) C (0011011111101011011011111011110011) C (1111000101100111100010101010000001) C (0011000110010100110010000110000001) C (1110101001000010010100101000011100) C (0110111010011110110001011011101001) C (1001100101111110111101011001101110) C (0000100111111101000010110011001001) C (1011001000010010011100101011000100) C (0101111110001111110000111111111011) C (0110101010100001110101011010110110) C (0001111000110101011111001111101111) C (1111011101111110000011100111111011) C (1101010100000100000001110100010001) C (0011110101110001000001111001110000) C (1100011110110111110101000101011111) C (0101011001011011111001010100001000) C (0000011110101100110001010101100011) C (1110100101011111001101011011011110) C (0001111000001111100010100011011010) C (1110011000000010100101000101010110) C (0100011111001011000000000010000011) Table 6.
Neighbours of N (4) C i ( x , x , ..., x ) γ β C i ( x , x , ..., x ) γ β C (0110011110100110101110111111001110) C (1010011101011110101011111111011110) C (1001110111100010010000100001111010) C (0011101011111100101001110010011011) C (1101101101010111100010000101001101) C (1001001100010110100011110011101101) C (0110000111110011101010000111110111) C (1110000100101011001100000100001101) C (1110111101111011001110111111010111) C (0111001000001101101001110011010010) C (0010010100101110111101101011111111) C (1101000000110100011000000110101000) C (0111101011001101101011010011001011) C (0001001011000000110111110000010110) C (1011011001100001001110011100101101) C (1000000111000110111000100010000001) C (1111010100111110101110110000011111) C (1000100000000111110001110000010010) C (0111011010010110011110110001000110) C (0010011100001000000010001111011000) C (0001101100111010100010011110101000) C (1000011011111111111010001110010001) C (0111010111111001111101011000101110) C (0101101101001100001001110011010010) C (1111011011111111111010100100111001) C (0011111100000101110110110111011111) C (0100111111101001101001110001101011) C (1111101111000110001100111111101100) C (0101111110001110100001110110011011) C (0111110111111110111101000001110100) C (1110110111101011000110100111111100) C (0000000111010010100010010001011001) C (0001001101110101011111001000101101) C (0100001111001011001010000111010011) C (0101000110111111010111000111000100) C (0110110111011011011110111101001100) C (0001110001110001001001110010111010) C (0001010100001110010110011101111101) C (0101010110001011110111000001101110) C (0010011111011010100011110101011011) C (0111100100111001111101100111110101) C (1101110110110011011001111111011011) C (1101011010110011000111101000101100) C (0110101110111110101011011111101011) C (1110000011001101000110000000101110) C (1001111010110000000101110100000100) C (1001000111100111010011111100111001) C (1011111011110111101111011111011100) C (0011111100110101110101101110110101) C (1011010011100011110000011000001011) C (0101001011001111001010011001000011) Conclusion
We introduced the concept of a distance between self-dual codes. This generalizes the notion of a neighborin self-dual codes, which leads to a new way of constructing new self-dual codes from a known one. Applyingthese ideas to an extremal binary self-dual code of length 68 we were able to construct 143 new extremal
ENERALIZED NEIGHBORS 7 binary self-dual codes of length 68 with new weight enumerators, including the first examples with γ = 8 , W , in the literature. Thus, we have now completed the theoretical list of possible γ values that can befound in W , . Generator matrices for some of the new codes are available online at [11]. 42 of the codeswe have constructed have γ = 8 in their weight enumerator, while 40 of them have γ = 9 in their weightenumerators. In particular, we have been able to construct the codes that have the following parameters:( γ = 5 , β = { , , , , } ) , ( γ = 6 , β = { , , , , , , , , , , , , , , , , , , , , , , , , , } ) , ( γ = 7 , β = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } ) , ( γ = 8 , β = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } ) , ( γ = 9 , β = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } )The strength of this new approach has been demonstrated by the number of new weight enumerators thatwe have been able to obtain by applying it to a single code. We believe this will open up new venues in thesearch and classification of new extremal binary self-dual codes. References [1] D. Anev, M. Harada and N. Yankov, “New extremal binary self-dual codes of lengths 64 and 66”,
J. Alg. Comb. Disc.Structures and Appl. , vol. 5, no.3, pp. 143–151, 2018.[2] W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system. I. The user language”,
J. Symbolic Comput ., Vol.24, pp. 235–265, 1997.[3] R.A. Brualdi and V.S. Pless, “Weight Enumerators of Self-Dual Codes”,
IEEE Trans. Infrom. Theory , vol. 37, no. 4, pp.1222-1225, 1991.[4] S. Buyuklıeva, I. Bouklıev, “Extremal self-dual codes with an automorphism of order 2”,
IEEE Trans. Inform. Theory ,Vol. 44, pp. 323–328, 1998.[5] J.H. Conway, N.J.A. Sloane, “A new upper bound on the minimal distance of self-dual codes”,
IEEE Trans. Inform.Theory , Vol. 36, 6, 1319–1333, 1990.[6] S.T. Dougherty, T.A. Gulliver, M. Harada,“Extremal binary self dual codes”,
IEEE Trans. Inform. Theory , Vol. 43 pp.2036-2047, 1997.[7] J. Gildea, H. Hamilton, A. Kaya and B. Yildiz, “Modified quadratic residue constructions and new extremal binary self-dualcodes of lengths 64, 66 and 68”,
Inf. Proces. Letters , vol. 157, 2020.[8] J. Gildea., A. Kaya, R. Taylor, and B. Yildiz, “Constructions for self-dual codes induced from group rings”,
Finite FieldsAppl. , vol. 51, pp. 71–92, 2018.[9] J. Gildea, A. Korban and A. Kaya, “Self-dual codes using bisymmetric designs and group rings”, submitted .[10] J. Gildea, A. Kaya and B. Yildiz, “An altered four circulant construction for self-dual codes from group rings and newextremal binary self-dual codes I”,
Discrete Mathematics. , vol. 342, no. 12, 111620, 2019.[11] J. Gildea, A. Kaya, A. Korban, B. Yildiz, “Binary generator matrices for some new extremal binary self-dual codes oflength 68”, available online at http://abidinkaya.wixsite.com/math/neighbor .[12] M. Harada, A. Munemasa, “Some restrictions on weight enumerators of singly even self-dual codes”,
IEEE Trans. Inform.Theory , Vol. 52, pp. 1266–1269, 2006.[13] A. Kaya and B. Yildiz “Various constructions for self-dual codes over rings and new binary self-dual codes”,
Discrete Math ,Vol. 339, No. 2 pp. 460–469, 2016.[14] A. Kaya, B. Yildiz, A. Pa¸sa “New extremal binary self-dual codes from a modified four circulant construction”,
DiscreteMath , Vol. 339, No. 3 pp. 1086–1094, 2016.[15] E.M. Rains, “Shadow Bounds for Self Dual Codes”,
IEEE Trans. Inf. Theory , Vol.44, pp.134–139, 1998.
Department of Mathematical and Physical Sciences, Faculty of Science and Engineering, University of Chester,England, UK
E-mail address : [email protected] JOE GILDEA, ABIDIN KAYA, ADRIAN KORBAN AND BAHATTIN YILDIZ
Department of Mathematics Education, Sampoerna University, 12780, Jakarta, Indonesia
E-mail address : [email protected] Department of Mathematical and Physical Sciences, Faculty of Science and Engineering, University of Chester,England, UK
E-mail address : [email protected] Department of Mathematics & Statistics, Northern Arizona University, Flagstaff, AZ 86001, USA
E-mail address ::