scholarly journals DUAL CODE AND MACWILLIAMS IDENTITY ON ADDITIVE CODE

2019 ◽  
Vol 1 (1) ◽  
Author(s):  
Miftah Yuliati ◽  
Sri Wahyuni ◽  
Indah Emilia Wijayanti
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Hamming Distance ◽  
Finite Abelian Group ◽  
Dual Code ◽  
Inner Product ◽  
Dual Codes ◽  
Macwilliams Identity ◽  
Homogeneous Weight ◽  
Distance Hamming

Additive code is a generalization of linear code. It is defined as subgroup of a finite Abelian group. The definitions of Hamming distance, Hamming weight, weight distribution, and homogeneous weight distribution in additive code are similar with the definitions in linear code. Different with linear code where the dual code is defined using inner product, additive code using theories in group to define its dual code because in group theory we do not have term of inner product. So, by this thesis, the definitions of dual code in additive code will be discussed. Then, this thesis discuss about a familiar theorem in dual code theory, that is MacWilliams Identity. Next, this thesis discuss about how to proof of MacWilliams Identity on adiitive code using dual codes which are defined.

NOTE ON SUPPORT WEIGHT DISTRIBUTION OF LINEAR CODES OVER

2015 ◽  
Vol 91 (2) ◽  
pp. 345-350 ◽  
Author(s):  
JIAN GAO
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Linear Codes ◽  
Dual Code ◽  
Support Weight

AbstractLet $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$, where $u^{2}=u$. A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{2k}$ over $R$ and its dual code $\mathscr{C}^{\bot }$ is established.

Linear codes over 𝔽4R and their MacWilliams identity

2020 ◽  
Vol 12 (06) ◽  
pp. 2050085
Author(s):  
Nasreddine Benbelkacem ◽  
Martianus Frederic Ezerman ◽  
Taher Abualrub
Keyword(s):  
Commutative Ring ◽  
Linear Codes ◽  
Inner Product ◽  
Dual Codes ◽  
Macwilliams Identity ◽  
Generator Matrices

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].

ON THE SUPPORT WEIGHT DISTRIBUTION OF LINEAR CODES OVER THE RING

2016 ◽  
Vol 95 (1) ◽  
pp. 157-163 ◽  
Author(s):  
MINJIA SHI ◽  
JIAQI FENG ◽  
JIAN GAO ◽  
ADEL ALAHMADI ◽  
PATRICK SOLÉ
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Linear Codes ◽  
Dual Code ◽  
Support Weight

Let $R=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+\cdots +u^{d-1}\mathbb{F}_{p}$, where $u^{d}=u$ and $p$ is a prime with $d-1$ dividing $p-1$. A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{dk}$ over $R$ and the dual code $\mathscr{C}^{\bot }$ is established.

On some punctured codes of ℤq-Simplex codes

2021 ◽  
pp. 2250012
Author(s):  
J. Prabu ◽  
J. Mahalakshmi ◽  
C. Durairajan ◽  
S. Santhakumar
Keyword(s):  
Prime Divisor ◽  
Weight Distribution ◽  
Linear Code ◽  
Prime Power ◽  
Punctured Codes ◽  
New Codes ◽  
Simplex Codes ◽  
Simplex Code ◽  
Partial Weight ◽  
Unit Formula

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

scholarly journals Nonexistence of Certain Singly Even Self-Dual Codes with Minimal Shadow

10.37236/7155 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Stefka Bouyuklieva ◽  
Masaaki Harada ◽  
Akihiro Munemasa
Keyword(s):  
Minimum Weight ◽  
Dual Code ◽  
Weight Enumerator ◽  
Dual Codes

It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even self-dual codes with minimal shadow having minimum weight $d-2$ for these $(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ and $m \ge 160$, respectively. We demonstrate that the weight enumerator of a singly even self-dual code with minimal shadow is not uniquely determined for parameters $[24m+6,12m+3,4m+2]$ and $[24m+22,12m+11,4m+4]$.

scholarly journals Implementation of a Hamming distance–like genomic quantum classifier using inner products on ibmqx2 and ibmq_16_melbourne

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Kunal Kathuria ◽  
Aakrosh Ratan ◽  
Michael McConnell ◽  
Stefan Bekiranov
Keyword(s):  
Hamming Distance ◽  
Training Sample ◽  
General Purpose ◽  
Inner Product ◽  
Matching Problems ◽  
Binary Operations ◽  
Inner Products ◽  
Training Class ◽  
Dot Product ◽  
Product Measures

Abstract Motivated by the problem of classifying individuals with a disease versus controls using a functional genomic attribute as input, we present relatively efficient general purpose inner product–based kernel classifiers to classify the test as a normal or disease sample. We encode each training sample as a string of 1 s (presence) and 0 s (absence) representing the attribute’s existence across ordered physical blocks of the subdivided genome. Having binary-valued features allows for highly efficient data encoding in the computational basis for classifiers relying on binary operations. Given that a natural distance between binary strings is Hamming distance, which shares properties with bit-string inner products, our two classifiers apply different inner product measures for classification. The active inner product (AIP) is a direct dot product–based classifier whereas the symmetric inner product (SIP) classifies upon scoring correspondingly matching genomic attributes. SIP is a strongly Hamming distance–based classifier generally applicable to binary attribute-matching problems whereas AIP has general applications as a simple dot product–based classifier. The classifiers implement an inner product between N = 2n dimension test and train vectors using n Fredkin gates while the training sets are respectively entangled with the class-label qubit, without use of an ancilla. Moreover, each training class can be composed of an arbitrary number m of samples that can be classically summed into one input string to effectively execute all test–train inner products simultaneously. Thus, our circuits require the same number of qubits for any number of training samples and are $O(\log {N})$ O ( log N ) in gate complexity after the states are prepared. Our classifiers were implemented on ibmqx2 (IBM-Q-team 2019b) and ibmq_16_melbourne (IBM-Q-team 2019a). The latter allowed encoding of 64 training features across the genome.

On the ℤq-Simplex codes and its weight distribution for dimension 2

2015 ◽  
Vol 07 (03) ◽  
pp. 1550030
Author(s):  
C. Durairajan ◽  
J. Mahalakshmi ◽  
P. Chella Pandian
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Prime Power ◽  
Dimension 2 ◽  
New Codes ◽  
Simplex Codes ◽  
Order Element

In this paper, we have defined ℤq-linear code and constructed some new codes. In particular, we have introduced the concept of ℤq-Simplex codes and proved that it is a [Formula: see text]-linear code for any integer q ≥ 2 and k ≥ 3 where p is the least order element in ℤq. We have given the weight distribution of ℤq-Simplex codes of dimension 2 when q is a prime power and when q is a product of distinct primes.

scholarly journals Studi Perbandingan Algoritma Pencarian String dalam Metode Approximate String Matching untuk Identifikasi Kesalahan Pengetikan Teks

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Yeny Rochmawati ◽  
Retno Kusumaningrum
Keyword(s):  
Hamming Distance ◽  
String Matching ◽  
Mean Average Precision ◽  
Levenshtein Distance ◽  
Approximate String Matching ◽  
Average Precision ◽  
Relevance Judgments ◽  
Typing Error ◽  
The Mean ◽  
Distance Hamming

Abstract. Error typing resulting in the change of standard words into non-standard words are often caused by misspelling. This can be addressed by developing a system to identify errors in typing. Approximate string matching is one method that is widely implemented to identify error typing by using several string search algorithms, i.e. Levenshtein Distance, Hamming Distance, Damerau Levenshtein Distance and Jaro Winkler Distance. However, there is no study that compares the performance of the four algorithms.  Therefore, this research aims to compare the performance between the four algorithms in order to identify which algorithm is the most accurate and precise in the search string based on various errors typing. Evaluation is performed by using users’ relevance judgments which produce the mean average precision (MAP) to determine the best algorithm. The result shows that Jaro Winkler Distance algorithm is the best in word-checking with 0.87 of MAP value when identifying the typing error of 50 incorrect words.Keywords: Errors typing, Levenshtein, Hamming, Damerau Levenshtein, Jaro Winkler Abstrak. Kesalahan pengetikan mengakibatkan kata baku berubah menjadi kata tidak baku karena ejaan yang digunakan tidak sesuai. Hal tersebut dapat ditangani dengan mengembangkan sistem untuk mengidentifikasi kesalahan pengetikan. Metode approximate string matching merupakan salah satu metode yang banyak diterapkan untuk mengidentifikasi kesalahan pengetikan dengan berbagai jenis algoritma pencarian string yaitu Levenshtein Distance, Hamming Distance, Damerau Levenshtein Distance dan Jaro Winkler Distance. Akan tetapi studi perbandingan kinerja dari keempat algoritma tersebut untuk Bahasa Indonesia belum pernah dilakukan. Oleh karena itu penelitian ini bertujuan untuk melakukan studi perbandingan kinerja dari keempat algoritma tersebut sehingga dapat diketahui algoritma mana yang lebih akurat dan tepat dalam pencarian string berdasarkan kesalahan penulisan yang bervariasi. Evaluasi yang dilakukan menggunakan user relevance judgement yang menghasilkan nilai mean average precision (MAP) untuk menentukan algoritma yang terbaik. Hasil penelitian terhadap 50 kata salah menunjukkan bahwa algoritma Jaro Winkler Distance terbaik dalam melakukan pengecekan kata dengan nilai MAP sebesar 0,87.Kata Kunci: Kesalahan pengetikan, Levenshtein, Hamming, Damerau Levenshtein, Jaro Winkler

scholarly journals Weight Distribution of Some Codewords of 3-ary Linear Code over GF(27)‎

2020 ◽  
Vol 31 (4) ◽  
pp. 101
Author(s):  
Maha Majeed Ibrahim ◽  
Emad Bakr Al-Zangana
Keyword(s):  
Weight Distribution ◽  
Linear Code ◽  
Linear Codes ◽  
Generator Matrix ◽  
Weight Distributions

This paper is devoted to introduce the structure of the p-ary linear codes C(n,q) of points and lines of PG(n,q),q=p^h prime. When p=3, the linear code C(2,27) is given with its generator matrix and also, some of weight distributions are calculated.

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