A REVERSIBLE CODE OVER GE(q)
|
|
- Philip Roberts
- 5 years ago
- Views:
Transcription
1 KYBERNETIKA- VOLUME 22 (1986), NUMBER 1 A REVERSIBLE CODE OVER GE(q) SUNIL KUMAR MUTTOO*, SHANKAR LAL This paper deals with the construction of codes over GF(q), q prime, by annexing two triangular matrices, one upper triangular and the other lower triangular. The error-correction capabilities of such codes are also studied. 0. INTRODUCTION An (n, ) linear code C of length n over GF(q) = F q, a Galios field of order q, where q is a prime, is a fc-dimensional linear subspace of F" q, where F" q denotes the space of all n-tuples over GF(q). A generator matrix G of this code is a x n matrix whose rows form a basis of C. The parity-chec matrix H of this code is an(n fc)x n matrix such that Hv T = 0 for all vectors vec. The row space of (n - ) x n matrix H is an (n, n fc) linear code C 1 called dual code of C. The Hamming weight of a vector is the number of non-zero elements in it. A code word in C consists of some fc symbols as message or information symbols and the remaining symbols as chec-digits [2]. A class of codes called 'reversible codes' introduced by Massey [3] is defined as follows: Definition. A linear code C is called reversible if a vector obtained by reversing the order of the digits of a code word in C result in a code word in C, i.e., (v 0, v u......, tf _i)e C implies that (v B t, v _ 2,..., v u v 0) e C. Consider a (fc + 1) x (2 + 1) matrix H over GF(q) (q prime) formed by annexing two square triangular matrices, one upper triangular and the other lower triangular such that the last column of the first is the first column of the second, where the * The author carried out this research wor under a minor research project sponsored by U.G.C., India. 85
2 entries are chosen in a well-defined way viz. consider H of the type, < x t x 2 x 3... x - t x y x 2 x 3... *,<_! x y x x 3... x _ 1 x y x x t _i 0 n /c + l X ~ 1 X y X X ~ l X ~ x У x x ~í 0 У x x -i x ~2 x 3 x 2 " x ~2 x ъ x г x \l where y, x t e(l, 2,..., q 1) and (x,-, y) = 1 and (x ;, Xj) = 1 for i 4= j, i,j = = 1, 2,...,. The code obtained by considering H as the parity-chec matrix will be a (2fc + 1, fc) linear code C. It will be shown that the code C which is the null space of H turns out to be reversible, as defined by Massey [3] and for further wor one may refer to Tzeng and Hartmann [6]. It is also shown that such codes are capable of correcting a well-defined class of solid burst errors. Codes correcting solid bursts have also been studied by Shiva and Sheng [5]. In what follows, by a solid burst of length b we shall mean an n-tuple whose all the b non-zero components are among some b adjacent positions. 1. CHARACTERIZATION In the following theorem we prove that the code C is a reversible code. We first prove a lemma. Lemma 1. The number of non-zero components in any code word of C, in the first fc-positions and in the last fc-positions is same. Proof. Let h u h 2,..., h 2+1 denote the columns of H, h ; denoting the ith column. The formation of the last fc-columns of H may be stated as K+2 - oa+i + Mi! ; 2fc+i = «A+i + l>a K+x + i = «A+i + b t hi, i = 1,2,..., fc where 0 < a t = q I, 0 < b t _ q 1. Let there be a code word with 5 non-zero components at the ijth, j 2 th,..., i s th place
3 in the last positions. Then (1) h +1+il + h +1 + i h +1 + is = = (a h + a i a is ) h +1 + (bifa + b i2 h b t h s ) Case I. When a h + a h a is = 0. The R.H.S. of (1) is clearly a sum of exactly s columns from the first fc-columns, h t, h 2,..., h of H. Thus there are exactly s non-zero components in the first and in the last positions of the code word under discussion. Case II. When 0 < a h + a i a is ^ q - 1. In this case, the R.H.S. of (l) is a sum of the ( + l)th column h + 1 and exactly s columns from the first columns h u h 2,..., h of H showing that there are exactly s non-zero components in the first positions and the last positions. Note that we have proved more than what has been stated in the lemma in view of the following corollary: Corollary 2. A one-to-one correspondence between the non-zero components in the first positions and the non-zero components in the last positions exist, viz. the ith component 1 < i ^ is related to the ( + I + i')th component. Theorem 3. C is reversible. Proof. Let v = (v u v 2,..., v 2+1 ) be a code word of C. Then 2+l (2) Z v t hi = 0. j = i We have (3) h +1 + i = fla+i + M;, where ( 4 ) K 1 = K-i + i and ~ ai b - i+1 = a _ i + 1 i = 1,2,3,...,/c; a h Equivalently, b i egf(q). (5) hi = bt l h +1+i - (a i b7 i )h +1 = b - i+1 h i + a _ i+1 h +1. Then (2) gives (using (3)) Y^v.hi + v +1 h +1 + Zv +t + i h +l + i = 0, E V І Һ І + v + iһ + i + Z v +l + i ( ai h + ì + bihi) = 0, Z("; + V Ł +n-i) /г ; + Z(a i^ ř + v +1 )h +1 = 0. 87
4 Therefore, we must have (6) v, + b-fi +l + t = 0, I"ft+iti + v +i = 0> i - 1,2,..., fc. Consider the vector v = (v 2+1, v 2,..., v + 2, v +i,..., v 2, v x ). Then vh T = S = v 2+1 h x + v 2 h v + 3 h _ 1 + v + 2 h + + v +ih+i + v K + i + + v 3h 2 -i + v 2 h 2 + v 1 h 2+1. Using (5) and then (4), we get S = (b v v ) h (b. l v 2 + i> t -i) h (b x v v t ) h Dv> 2 *+i + a»-i»2* + a. 2 v 2 -i a t v +2 + v +1 ] h +x which on using (6) gives S = 0. Thus v is a code of C. Hence C is reversible. Remar. Taing a t 1, fe, = 1, i = 1, 2,..., fc, we have the relations lt*+i + i = K + i + >h. The codes generated by the matrix satisfying the above conditions in the binary case i.e. over GE(2) have been studied by Dass and Muttoo [1]. Theorem 4. The dual, C, of the reversible code C is reversible. Proof. A parity chec matrix of the code C is a generator matrix for the code C. The code words of C are various linear combinations of the rows of H and the rows of H are such that the reverse of any row of H is again a row of H. Therefore, the reverse of a code word of C is a code word of C. This completes the proof of the theorem. 2. ERROR CORRECTION CAPABILITIES The following result determines the error-correction capabilities of the reversible codes considered in this paper. Theorem 5. An (n, fc) linear code C, n = 2fc + 1, whose parity chec matrix is H, is capable to correct, (i) all solids bursts of odd lengths upto 2fc 1, if n is even, i.e. iffcis odd, (ii) all solids bursts of odd lengths upto fc 1, if n is odd, i.e. if fc is even. Proof. Firstly, we shall derive an upper bound on the sufficient number of paritychec digits for the existence of a code that is capable to correct a solid burst of odd length, say b, by constructing a suitable parity-chec matrix. The procedure involves
5 suitable modifications of the technique used in deriving Varshamov-Gilbert-Sacs bound [4]. After having selected the first (j - l) columns h t, h 2,, h j_ 1 of the parity-chec matrix, the jth column hj to be added to the matrix should be such that hj * ( aj_ b+1hj_ b+1 + aj b + 2hj_ b O-ifcy-i) + Mi + + b i+1h i b i+b_ 1h i+ b_ 1, where the columns h t are any fo-consecutive columns among the first (j i l) columns, / = 0, 2, 4,..., b 1, and all the coefficients a t and b t are non-zero. Thus, the coefficients a ; form a solid burst of even length b 1 or less and the coefficients fr ; form a solid burst of odd length b or less in a (j i l)-tuple. The number of choices of these coefficients can be calculated as follows: If a solid burst of even length is of length (b l) then the number of solid bursts of odd length b or less in a (j - b) - tuple is (Vx) (j -b)(q-l) + (j-b~ 2)(q - l) (j - b - b + l)(fl - l) 6. If a solid burst of even length is of length (b - 3) then the number of solid bursts of odd length b or less in a (j - b + 2)-tuple is (Va) (j - b + 2)(q - 1) + (; - b)(q - l) (j - b b + l)(q - l) b If the solid burst of even length is of length zero, then the number of solid bursts of odd length b or less in a (j - l)-tuple is (7 0) (J - 1)(«- 1) + (j - 3)(a - l) (j -b)(q~ 1)". Therefore, the total number of possible choices of the coefficients a ; andfo ; are («-l) d - 1 (7 i,- 1) + (a-l) 6-3 (7 i,- 3) (7o) which on simplification gives (»+J)/2 (&-l)/2 (6-l)/2 (8) [ I (J - 2i + l)(q~ If" 1 - X q{q - l) 2; + 1 ] (q - If. i=i i=i ;=o At worst, if all the linear combinations yield a distinct sum, then the y'th column hj can be added to H if (9) <r*>(8). But for an (n, ) linear code to exist, the inequality in (9) should hold for j = n, and we get (6+l)/2 (h-d/2 (10) q"' > [ X ( _ 2i + l)(q ~ If 1-2i(q - if +1 ]. ; = i (6-l)/2 Z (4-i) 2; - i = 0 We now prove the main result.
6 Case (i). Let be odd. The solid bursts of odd lengths upto 2 1 are the solid bursts of lengths 2 1,2 3,..., 3, 1. For these values of b, the inequality in (10) has the form (11) «* +1 >Pl'-4,-2"l 1 -»J[TcJ where A ; = ( - i + l) (q - l) 2i ~ 1,B i = ife - l) 2;+1, C, = (_ - l) 2 '', s = 0, 1,2,..., i. To prove the above claim, we employ induction technique. We wish to prove that Now -s+2 -s+1 -s+1 q + 3 > [2 A ;] [ _L Cj. ; = i ^+i >[ 2 'xa ; -2"i: 1 JB ; ][T 1 c;] = i=i = [ - 2(A,_ S+1 + A,_ s+2 - _?*_ s - 5,_ s+1)] [Y- C,_ s - C,_ s + 1] fc-s+2 -s+l -s+l where X = 2 A; - 2 _? ;, 7= C,, Thus ; = l ; = l ; = i q*+i > [XY-X(C _. + C,_ s+1) - 2F(A,_ S+1 + A,_ s+2 - B _ s - B,_ s+1) + + 2(C,_ S + C,_ S+1)(A,_ S+1 + A,_ s+2 - B _ s - +_)]. As s ; 1, _ 1, therefore - fc+3 > ^+1 + _qc + c&s+i) + 2Y(A,_ S+1 + A,_ s f?,_ s - B,_ s+]) - - 2(C,_ S + C,_ S+1)(A,_ S+1 + A,_ s+2-5,_ s - B t+1) > XY. Thus the inequality in (10) is true for all odd values of. Hence the case (i). Case (ii). Let be even. The solid bursts of odd lengths up to are the bursts of lengthsfe 1, 3,......, 5, 3, 1. For these values of b the bound in (10) has the form (t-2s)/2 (-2s-2)/2 (-2s-2)/2 (12) g" +1 >[2 A,-2 5 ;] C ;, s = 0,l,2,...,(/c-2)/2, ; = i ; = o where A ;, B t and C ; are as in (11). To prove that this is true, we shall use the induction technique. We wish to prove that (*:-2s+2)/2 (-2s)/2 (-2s)/2 q + 3 >[2 _Z A ; ~2 _?,] X C ;. i=o Using the technique of case (i), the result follows. 90
7 Example. Consider the following (4 x 7) matrix H 3 over GF(5): Я, = The columns of this matrix satisfy the relations where K+i + t = aa+i + b A, i =1,2,3 a 1 = 2, b l = 2, a 2 = 2, b 2 = 4, a 3 4, b 3 = 3. It can be seen that the null space of H 3, is a reversible code and corrects all solid bursts of lengths 1, 3, 5. ACKNOWLEDGEMENT The authors are thanful to Dr. B. K. Dass, Department of Mathematics, P. G. D. A. V. College, (University of Delhi), for his fruitful suggestions. REFERENCES (Received June 14, 1984.) [1] B. K. Dass and S. K. Muttoo: A reversible code and its characterization. Presented at the 46th Conference of the Indian Mathematical Society held at Bangalore, December 27 29, [2] F. J. Mac Williams and N. J. A. Sloane: The Theory of Error Correcting Codes. North Holland, Amsterdam [3] J. L. Massey: Reversible Codes. Inform, and Control 7 (1964), [4] W. W. Peterson and E. J. Weldon, Jr.: Error Correcting Codes. Second edition. MIT Press, Cambridge, Mass [5] S. G. S. Shiva and C. L. Sheng: Multiph solid burst-error-correcting binary codes. IEEE Trans. Inform. Theory IT-15 (1969), [6] K. K. Tzeng and C. R. P. Hartmann: On minimum distance of certain reversible cyclx codes. IEEE Trans. Inform. Theory IT-16 (1970), Dr. Sunil Kumar Muttoo, Department of Mathematics, S. G. T. B. Khasa College, University of Delhi, Delhi India. Shanar Lai, Department of Mathematics, Zair Hussain College (E), University of Delhi, Delhi India. 91
Plotkin s Bound in Codes Equipped with the Euclidean Weight Function
Tamsui Oxford Journal of Mathematical Sciences 5() (009) 07-4 Aletheia University Plotkin s Bound in Codes Equipped with the Euclidean Weight Function Sapna Jain Department of Mathematics, University of
More informationConstruction of m-repeated Burst Error Detecting and Correcting Non-binary Linear Codes
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 1 (June 2017), pp. 337-349 Applications and Applied Mathematics: An International Journal (AAM) Construction of m-repeated
More informationA Relation Between Weight Enumerating Function and Number of Full Rank Sub-matrices
A Relation Between Weight Enumerating Function and Number of Full Ran Sub-matrices Mahesh Babu Vaddi and B Sundar Rajan Department of Electrical Communication Engineering, Indian Institute of Science,
More informationA Note on Non-Existence of (8,4)-2 Burst Error Correcting Code
Applied Mathematical Sciences, Vol. 4, 2010, no. 44, 2201-2205 A Note on Non-Existence of (8,4)-2 Burst Error Correcting Code Navneet Singh Rana Research Scholar, Department of Mathematics University of
More informationAn Application of Coding Theory into Experimental Design Construction Methods for Unequal Orthogonal Arrays
The 2006 International Seminar of E-commerce Academic and Application Research Tainan, Taiwan, R.O.C, March 1-2, 2006 An Application of Coding Theory into Experimental Design Construction Methods for Unequal
More informationON PERTURBATION OF BINARY LINEAR CODES. PANKAJ K. DAS and LALIT K. VASHISHT
Math Appl 4 (2015), 91 99 DOI: 1013164/ma201507 ON PERTURBATION OF BINARY LINEAR CODES PANKAJ K DAS and LALIT K VASHISHT Abstract We present new codes by perturbation of rows of the generating matrix of
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationCharacterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs. Sung Y. Song Iowa State University
Characterizations of Strongly Regular Graphs: Part II: Bose-Mesner algebras of graphs Sung Y. Song Iowa State University sysong@iastate.edu Notation K: one of the fields R or C X: a nonempty finite set
More informationOn Two Probabilistic Decoding Algorithms for Binary Linear Codes
On Two Probabilistic Decoding Algorithms for Binary Linear Codes Miodrag Živković Abstract A generalization of Sullivan inequality on the ratio of the probability of a linear code to that of any of its
More informationCyclic codes: overview
Cyclic codes: overview EE 387, Notes 14, Handout #22 A linear block code is cyclic if the cyclic shift of a codeword is a codeword. Cyclic codes have many advantages. Elegant algebraic descriptions: c(x)
More informationOn the construction of asymmetric orthogonal arrays
isid/ms/2015/03 March 05, 2015 http://wwwisidacin/ statmath/indexphp?module=preprint On the construction of asymmetric orthogonal arrays Tianfang Zhang and Aloke Dey Indian Statistical Institute, Delhi
More informationSolutions of Exam Coding Theory (2MMC30), 23 June (1.a) Consider the 4 4 matrices as words in F 16
Solutions of Exam Coding Theory (2MMC30), 23 June 2016 (1.a) Consider the 4 4 matrices as words in F 16 2, the binary vector space of dimension 16. C is the code of all binary 4 4 matrices such that the
More informationELEC 519A Selected Topics in Digital Communications: Information Theory. Hamming Codes and Bounds on Codes
ELEC 519A Selected Topics in Digital Communications: Information Theory Hamming Codes and Bounds on Codes Single Error Correcting Codes 2 Hamming Codes (7,4,3) Hamming code 1 0 0 0 0 1 1 0 1 0 0 1 0 1
More informationVector spaces. EE 387, Notes 8, Handout #12
Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is
More informationPermutation decoding for the binary codes from triangular graphs
Permutation decoding for the binary codes from triangular graphs J. D. Key J. Moori B. G. Rodrigues August 6, 2003 Abstract By finding explicit PD-sets we show that permutation decoding can be used for
More informationWEIGHT DISTRIBUTIONS OF SOME CLASSES OF BINARY CYCLIC CODES
Syracuse University SURFACE Electrical Engineering and Computer Science Technical Reports College of Engineering and Computer Science 3-1974 WEIGHT DISTRIBUTIONS OF SOME CLASSES OF BINARY CYCLIC CODES
More informationLecture 17: Perfect Codes and Gilbert-Varshamov Bound
Lecture 17: Perfect Codes and Gilbert-Varshamov Bound Maximality of Hamming code Lemma Let C be a code with distance 3, then: C 2n n + 1 Codes that meet this bound: Perfect codes Hamming code is a perfect
More informationChapter 3 Linear Block Codes
Wireless Information Transmission System Lab. Chapter 3 Linear Block Codes Institute of Communications Engineering National Sun Yat-sen University Outlines Introduction to linear block codes Syndrome and
More informationAffine designs and linear orthogonal arrays
Affine designs and linear orthogonal arrays Vladimir D. Tonchev Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA, tonchev@mtu.edu Abstract It is proved
More informationICT12 8. Linear codes. The Gilbert-Varshamov lower bound and the MacWilliams identities SXD
1 ICT12 8. Linear codes. The Gilbert-Varshamov lower bound and the MacWilliams identities 19.10.2012 SXD 8.1. The Gilbert Varshamov existence condition 8.2. The MacWilliams identities 2 8.1. The Gilbert
More informationSPRING OF 2008 D. DETERMINANTS
18024 SPRING OF 2008 D DETERMINANTS In many applications of linear algebra to calculus and geometry, the concept of a determinant plays an important role This chapter studies the basic properties of determinants
More informationQuantum LDPC Codes Derived from Combinatorial Objects and Latin Squares
Codes Derived from Combinatorial Objects and s Salah A. Aly & Latin salah at cs.tamu.edu PhD Candidate Department of Computer Science Texas A&M University November 11, 2007 Motivation for Computers computers
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationERROR-CORRECTING CODES AND LATIN SQUARES
ERROR-CORRECTING CODES AND LATIN SQUARES Ritu Ahuja Department of Mathematics, Khalsa College for Women, Civil Lines, Ludhiana 141001, Punjab, (India) ABSTRACT Data stored and transmitted in digital form
More informationThe Hamming Codes and Delsarte s Linear Programming Bound
The Hamming Codes and Delsarte s Linear Programming Bound by Sky McKinley Under the Astute Tutelage of Professor John S. Caughman, IV A thesis submitted in partial fulfillment of the requirements for the
More informationA Questionable Distance-Regular Graph
A Questionable Distance-Regular Graph Rebecca Ross Abstract In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection
More informationx n k m(x) ) Codewords can be characterized by (and errors detected by): c(x) mod g(x) = 0 c(x)h(x) = 0 mod (x n 1)
Cyclic codes: review EE 387, Notes 15, Handout #26 A cyclic code is a LBC such that every cyclic shift of a codeword is a codeword. A cyclic code has generator polynomial g(x) that is a divisor of every
More informationOrthogonal Arrays & Codes
Orthogonal Arrays & Codes Orthogonal Arrays - Redux An orthogonal array of strength t, a t-(v,k,λ)-oa, is a λv t x k array of v symbols, such that in any t columns of the array every one of the possible
More informationSPHERE PACKINGS CONSTRUCTED FROM BCH AND JUSTESEN CODES
SPHERE PACKINGS CONSTRUCTED FROM BCH AND JUSTESEN CODES N. J. A. SLOANE Abstract. Bose-Chaudhuri-Hocquenghem and Justesen codes are used to pack equa spheres in M-dimensional Euclidean space with density
More informationIdeals over a Non-Commutative Ring and their Application in Cryptology
Ideals over a Non-Commutative Ring and their Application in Cryptology E. M. Gabidulin, A. V. Paramonov and 0. V. Tretjakov Moscow Institute of Physics and Technology 141700 Dolgoprudnii Moscow Region,
More informationStructured Low-Density Parity-Check Codes: Algebraic Constructions
Structured Low-Density Parity-Check Codes: Algebraic Constructions Shu Lin Department of Electrical and Computer Engineering University of California, Davis Davis, California 95616 Email:shulin@ece.ucdavis.edu
More informationAlternant and BCH codes over certain rings
Computational and Applied Mathematics Vol. 22, N. 2, pp. 233 247, 2003 Copyright 2003 SBMAC Alternant and BCH codes over certain rings A.A. ANDRADE 1, J.C. INTERLANDO 1 and R. PALAZZO JR. 2 1 Department
More informationRON M. ROTH * GADIEL SEROUSSI **
ENCODING AND DECODING OF BCH CODES USING LIGHT AND SHORT CODEWORDS RON M. ROTH * AND GADIEL SEROUSSI ** ABSTRACT It is shown that every q-ary primitive BCH code of designed distance δ and sufficiently
More informationMatrix-Product Complementary dual Codes
Matrix-Product Complementary dual Codes arxiv:1604.03774v1 [cs.it] 13 Apr 2016 Xiusheng Liu School of Mathematics and Physics, Hubei Polytechnic University Huangshi, Hubei 435003, China, Email: lxs6682@163.com
More informationM 2 + s 2. Note that the required matrix A when M 2 + s 2 was also obtained earlier by Gordon [2]. (2.2) x -alxn-l-aex n-2 an
SIAM J. ALG. DISC. METH. Vol. 1, No. 1, March 1980 1980 Society for. Industrial and Applied Mathematics 0196-52/80/0101-0014 $01.00/0 ON CONSTRUCTION OF MATRICES WITH DISTINCT SUBMATRICES* SHARAD V. KANETKAR"
More informationPAijpam.eu CONVOLUTIONAL CODES DERIVED FROM MELAS CODES
International Journal of Pure and Applied Mathematics Volume 85 No. 6 013, 1001-1008 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v85i6.3
More informationOn the adjacency matrix of a block graph
On the adjacency matrix of a block graph R. B. Bapat Stat-Math Unit Indian Statistical Institute, Delhi 7-SJSS Marg, New Delhi 110 016, India. email: rbb@isid.ac.in Souvik Roy Economics and Planning Unit
More informationConstruction of some new families of nested orthogonal arrays
isid/ms/2017/01 April 7, 2017 http://www.isid.ac.in/ statmath/index.php?module=preprint Construction of some new families of nested orthogonal arrays Tian-fang Zhang, Guobin Wu and Aloke Dey Indian Statistical
More informationarxiv: v1 [math.co] 27 Jul 2015
Perfect Graeco-Latin balanced incomplete block designs and related designs arxiv:1507.07336v1 [math.co] 27 Jul 2015 Sunanda Bagchi Theoretical Statistics and Mathematics Unit Indian Statistical Institute
More informationNecessary And Sufficient Conditions For Existence of the LU Factorization of an Arbitrary Matrix.
arxiv:math/0506382v1 [math.na] 19 Jun 2005 Necessary And Sufficient Conditions For Existence of the LU Factorization of an Arbitrary Matrix. Adviser: Charles R. Johnson Department of Mathematics College
More informationSecret-sharing with a class of ternary codes
Theoretical Computer Science 246 (2000) 285 298 www.elsevier.com/locate/tcs Note Secret-sharing with a class of ternary codes Cunsheng Ding a, David R Kohel b, San Ling c; a Department of Computer Science,
More informationInteraction balance in symmetrical factorial designs with generalized minimum aberration
Interaction balance in symmetrical factorial designs with generalized minimum aberration Mingyao Ai and Shuyuan He LMAM, School of Mathematical Sciences, Peing University, Beijing 100871, P. R. China Abstract:
More informationBOUNDS FOR HIGHER TOPOLOGICAL COMPLEXITY OF REAL PROJECTIVE SPACE IMPLIED BY BP
BOUNDS FOR HIGHER TOPOLOGICAL COMPLEXITY OF REAL PROJECTIVE SPACE IMPLIED BY BP DONALD M. DAVIS Abstract. We use Brown-Peterson cohomology to obtain lower bounds for the higher topological complexity,
More informationMath 321: Linear Algebra
Math 32: Linear Algebra T. Kapitula Department of Mathematics and Statistics University of New Mexico September 8, 24 Textbook: Linear Algebra,by J. Hefferon E-mail: kapitula@math.unm.edu Prof. Kapitula,
More informationA class of linear partition error control codes inγ-metric
World Applied Sciences Journal (Special Issue of Applied Math): 3-35, 3 ISSN 88-495 c IDOSI Publications, 3 DOI: 589/idosiwasjam53 A class of linear partition error control codes inγ-metric Sapna Jain
More informationMatrix characterization of linear codes with arbitrary Hamming weight hierarchy
Linear Algebra and its Applications 412 (2006) 396 407 www.elsevier.com/locate/laa Matrix characterization of linear codes with arbitrary Hamming weight hierarchy G. Viswanath, B. Sundar Rajan Department
More informationCLASS 12 ALGEBRA OF MATRICES
CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More informationP = 1 F m(p ) = IP = P I = f(i) = QI = IQ = 1 F m(p ) = Q, so we are done.
Section 1.6: Invertible Matrices One can show (exercise) that the composition of finitely many invertible functions is invertible. As a result, we have the following: Theorem 6.1: Any admissible row operation
More informationQUADRATIC RESIDUE CODES OVER Z 9
J. Korean Math. Soc. 46 (009), No. 1, pp. 13 30 QUADRATIC RESIDUE CODES OVER Z 9 Bijan Taeri Abstract. A subset of n tuples of elements of Z 9 is said to be a code over Z 9 if it is a Z 9 -module. In this
More informationMathematics Department
Mathematics Department Matthew Pressland Room 7.355 V57 WT 27/8 Advanced Higher Mathematics for INFOTECH Exercise Sheet 2. Let C F 6 3 be the linear code defined by the generator matrix G = 2 2 (a) Find
More informationarxiv: v2 [math.co] 17 May 2017
New nonbinary code bounds based on divisibility arguments Sven Polak 1 arxiv:1606.05144v [math.co] 17 May 017 Abstract. For q, n, d N, let A q (n, d) be the maximum size of a code C [q] n with minimum
More informationCIRCULAR CHROMATIC NUMBER AND GRAPH MINORS. Xuding Zhu 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 4, No. 4, pp. 643-660, December 2000 CIRCULAR CHROMATIC NUMBER AND GRAPH MINORS Xuding Zhu Abstract. This paper proves that for any integer n 4 and any rational number
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationLIFTED CODES OVER FINITE CHAIN RINGS
Math. J. Okayama Univ. 53 (2011), 39 53 LIFTED CODES OVER FINITE CHAIN RINGS Steven T. Dougherty, Hongwei Liu and Young Ho Park Abstract. In this paper, we study lifted codes over finite chain rings. We
More informationConstructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach
Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach Shu Lin, Shumei Song, Lan Lan, Lingqi Zeng and Ying Y Tai Department of Electrical & Computer Engineering University of California,
More informationIntroduction to Formal Languages, Automata and Computability p.1/51
Introduction to Formal Languages, Automata and Computability Finite State Automata K. Krithivasan and R. Rama Introduction to Formal Languages, Automata and Computability p.1/51 Introduction As another
More informationGreedy Codes. Theodore Rice
Greedy Codes Theodore Rice 1 Key ideas These greedy codes come from certain positions of combinatorial games. These codes will be identical to other well known codes. Little sophisticated mathematics is
More informationSums of diagonalizable matrices
Linear Algebra and its Applications 315 (2000) 1 23 www.elsevier.com/locate/laa Sums of diagonalizable matrices J.D. Botha Department of Mathematics University of South Africa P.O. Box 392 Pretoria 0003
More informationA Class of Quantum LDPC Codes Constructed From Finite Geometries
A Class of Quantum LDPC Codes Constructed From Finite Geometries Salah A Aly Department of Computer Science, Texas A&M University College Station, TX 77843, USA Email: salah@cstamuedu arxiv:07124115v3
More informationType I Codes over GF(4)
Type I Codes over GF(4) Hyun Kwang Kim San 31, Hyoja Dong Department of Mathematics Pohang University of Science and Technology Pohang, 790-784, Korea e-mail: hkkim@postech.ac.kr Dae Kyu Kim School of
More informationExtending MDS Codes. T. L. Alderson
Extending MDS Codes T. L. Alderson Abstract A q-ary (n,k)-mds code, linear or not, satisfies n q + k 1. A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show
More informationIdempotent Generators of Generalized Residue Codes
1 Idempotent Generators of Generalized Residue Codes A.J. van Zanten A.J.vanZanten@uvt.nl Department of Communication and Informatics, University of Tilburg, The Netherlands A. Bojilov a.t.bozhilov@uvt.nl,bojilov@fmi.uni-sofia.bg
More information3. Coding theory 3.1. Basic concepts
3. CODING THEORY 1 3. Coding theory 3.1. Basic concepts In this chapter we will discuss briefly some aspects of error correcting codes. The main problem is that if information is sent via a noisy channel,
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More informationChapter 6. BCH Codes
Chapter 6 BCH Codes Description of the Codes Decoding of the BCH Codes Outline Implementation of Galois Field Arithmetic Implementation of Error Correction Nonbinary BCH Codes and Reed-Solomon Codes Weight
More informationBinary Convolutional Codes of High Rate Øyvind Ytrehus
Binary Convolutional Codes of High Rate Øyvind Ytrehus Abstract The function N(r; ; d free ), defined as the maximum n such that there exists a binary convolutional code of block length n, dimension n
More informationREED-SOLOMON CODE SYMBOL AVOIDANCE
Vol105(1) March 2014 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 13 REED-SOLOMON CODE SYMBOL AVOIDANCE T Shongwe and A J Han Vinck Department of Electrical and Electronic Engineering Science, University
More informationOn the Complexity of the Dual Bases of the Gaussian Normal Bases
Algebra Colloquium 22 (Spec ) (205) 909 922 DOI: 0.42/S00538675000760 Algebra Colloquium c 205 AMSS CAS & SUZHOU UNIV On the Complexity of the Dual Bases of the Gaussian Normal Bases Algebra Colloq. 205.22:909-922.
More informationLectures 2 3 : Wigner s semicircle law
Fall 009 MATH 833 Random Matrices B. Való Lectures 3 : Wigner s semicircle law Notes prepared by: M. Koyama As we set up last wee, let M n = [X ij ] n i,j= be a symmetric n n matrix with Random entries
More informationOn Subsets with Cardinalities of Intersections Divisible by a Fixed Integer
Europ. J. Combinatorics (1983) 4,215-220 On Subsets with Cardinalities of Intersections Divisible by a Fixed Integer P. FRANKL AND A. M. ODLYZKO If m(n, I) denotes the maximum number of subsets of an n-element
More informationAn Exponential Number of Generalized Kerdock Codes
INFORMATION AND CONTROL 53, 74--80 (1982) An Exponential Number of Generalized Kerdock Codes WILLIAM M. KANTOR* Department of Mathematics, University of Oregon, Eugene, Oregon 97403 If n- 1 is an odd composite
More informationThe BCH Bound. Background. Parity Check Matrix for BCH Code. Minimum Distance of Cyclic Codes
S-723410 BCH and Reed-Solomon Codes 1 S-723410 BCH and Reed-Solomon Codes 3 Background The algebraic structure of linear codes and, in particular, cyclic linear codes, enables efficient encoding and decoding
More informationT linear programming to be applied successfully to error-correcting
1276 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 34, NO. 5, SEPTEMBER 1988 Inequalities for Covering Codes Abstract-Any code C with covering radius R must satisfy a set of linear inequalities that involve
More informationUpper triangular matrices and Billiard Arrays
Linear Algebra and its Applications 493 (2016) 508 536 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Upper triangular matrices and Billiard Arrays
More informationMTH6108 Coding theory
MTH6108 Coding theory Contents 1 Introduction and definitions 2 2 Good codes 6 2.1 The main coding theory problem............................ 6 2.2 The Singleton bound...................................
More informationLesson 3. Inverse of Matrices by Determinants and Gauss-Jordan Method
Module 1: Matrices and Linear Algebra Lesson 3 Inverse of Matrices by Determinants and Gauss-Jordan Method 3.1 Introduction In lecture 1 we have seen addition and multiplication of matrices. Here we shall
More informationCONSTRUCTION OF SLICED SPACE-FILLING DESIGNS BASED ON BALANCED SLICED ORTHOGONAL ARRAYS
Statistica Sinica 24 (2014), 1685-1702 doi:http://dx.doi.org/10.5705/ss.2013.239 CONSTRUCTION OF SLICED SPACE-FILLING DESIGNS BASED ON BALANCED SLICED ORTHOGONAL ARRAYS Mingyao Ai 1, Bochuan Jiang 1,2
More informationRepresentations of disjoint unions of complete graphs
Discrete Mathematics 307 (2007) 1191 1198 Note Representations of disjoint unions of complete graphs Anthony B. Evans Department of Mathematics and Statistics, Wright State University, Dayton, OH, USA
More informationHöst, Stefan; Johannesson, Rolf; Zigangirov, Kamil; Zyablov, Viktor V.
Active distances for convolutional codes Höst, Stefan; Johannesson, Rolf; Zigangirov, Kamil; Zyablov, Viktor V Published in: IEEE Transactions on Information Theory DOI: 101109/18749009 Published: 1999-01-01
More informationCOMPUTATIONAL COMPLEXITY OF PARAMETRIC LINEAR PROGRAMMING +
Mathematical Programming 19 (1980) 213-219. North-Holland Publishing Company COMPUTATIONAL COMPLEXITY OF PARAMETRIC LINEAR PROGRAMMING + Katta G. MURTY The University of Michigan, Ann Arbor, MI, U.S.A.
More informationConstruction of latin squares of prime order
Construction of latin squares of prime order Theorem. If p is prime, then there exist p 1 MOLS of order p. Construction: The elements in the latin square will be the elements of Z p, the integers modulo
More informationZero-sum square matrices
Zero-sum square matrices Paul Balister Yair Caro Cecil Rousseau Raphael Yuster Abstract Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the
More informationAN EXTENSION OF A THEOREM OF EULER. 1. Introduction
AN EXTENSION OF A THEOREM OF EULER NORIKO HIRATA-KOHNO, SHANTA LAISHRAM, T. N. SHOREY, AND R. TIJDEMAN Abstract. It is proved that equation (1 with 4 109 does not hold. The paper contains analogous result
More informationOn Construction of a Class of. Orthogonal Arrays
On Construction of a Class of Orthogonal Arrays arxiv:1210.6923v1 [cs.dm] 25 Oct 2012 by Ankit Pat under the esteemed guidance of Professor Somesh Kumar A Dissertation Submitted for the Partial Fulfillment
More informationMAS309 Coding theory
MAS309 Coding theory Matthew Fayers January March 2008 This is a set of notes which is supposed to augment your own notes for the Coding Theory course They were written by Matthew Fayers, and very lightly
More informationAverage Coset Weight Distributions of Gallager s LDPC Code Ensemble
1 Average Coset Weight Distributions of Gallager s LDPC Code Ensemble Tadashi Wadayama Abstract In this correspondence, the average coset eight distributions of Gallager s LDPC code ensemble are investigated.
More informationMATH 291T CODING THEORY
California State University, Fresno MATH 291T CODING THEORY Spring 2009 Instructor : Stefaan Delcroix Chapter 1 Introduction to Error-Correcting Codes It happens quite often that a message becomes corrupt
More informationCodes from lattice and related graphs, and permutation decoding
Codes from lattice and related graphs, and permutation decoding J. D. Key School of Mathematical Sciences University of KwaZulu-Natal Pietermaritzburg 3209, South Africa B. G. Rodrigues School of Mathematical
More informationGraph-based codes for flash memory
1/28 Graph-based codes for flash memory Discrete Mathematics Seminar September 3, 2013 Katie Haymaker Joint work with Professor Christine Kelley University of Nebraska-Lincoln 2/28 Outline 1 Background
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES
ARITHMETICALLY COHEN-MACAULAY BUNDLES ON THREE DIMENSIONAL HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle
More informationNew Traceability Codes against a Generalized Collusion Attack for Digital Fingerprinting
New Traceability Codes against a Generalized Collusion Attack for Digital Fingerprinting Hideki Yagi 1, Toshiyasu Matsushima 2, and Shigeichi Hirasawa 2 1 Media Network Center, Waseda University 1-6-1,
More informationON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY
ON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY YUFEI ZHAO Abstract. In this paper we discuss the Bruhat order of the symmetric group. We give two criteria for comparing elements in this
More information1 I A Q E B A I E Q 1 A ; E Q A I A (2) A : (3) A : (4)
Latin Squares Denition and examples Denition. (Latin Square) An n n Latin square, or a latin square of order n, is a square array with n symbols arranged so that each symbol appears just once in each row
More informationA TWOOA Construction for Multi-Receiver Multi-Message Authentication Codes
A TWOOA Construction for Multi-Receiver Multi-Message Authentication Codes R Fuji-Hara Graduate School of Systems and Information Engineering University of Tsukuba Tsukuba 305-8573, Japan X Li Department
More informationCh. 5 Determinants. Ring Determinant functions Existence, Uniqueness and Properties
Ch. 5 Determinants Ring Determinant functions Existence, Uniqueness and Properties Rings A ring is a set K with operations (x,y)->x+y. (x,y)->xy. (a) K is commutative under + (b) (xy)z=x(yz) (c ) x(y+z)=xy+xz,
More informationCourse MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography
Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups
More informationGeneric degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach
Generic degree structure of the minimal polynomial nullspace basis: a block Toeplitz matrix approach Bhaskar Ramasubramanian 1, Swanand R. Khare and Madhu N. Belur 3 December 17, 014 1 Department of Electrical
More informationA Piggybacking Design Framework for Read-and Download-efficient Distributed Storage Codes
A Piggybacing Design Framewor for Read-and Download-efficient Distributed Storage Codes K V Rashmi, Nihar B Shah, Kannan Ramchandran, Fellow, IEEE Department of Electrical Engineering and Computer Sciences
More information