distance d is the minimum weight among all the nonzero vectors in C, i.e., d= min
u∈Cu̸=0
w(u)
A linear code can be concisely specified with the help of agenerator matrix.
Definition 2.18. A generator matrix G of an [n, k, d] linear code C over Fq is any k×n matrix whose rows form a basis for C.
Note that ifG is a generator matrix for C then, C ={vG:v∈Fkq}.
We can also describe C with the help of an (n−k)×n matrix H, called parity check matrix, such that C is the null space of H, i.e.,
v∈C ⇐⇒ HvT = 0.
If the generator matrix G of a linear code C has the form G = [Ik|A], then we say that Gis instandard form. The following result shows the relationship between the generator matrix in standard form and the parity check matrix of a linear code C.
Theorem 2.26. If G= [Ik |A]is a generator matrix of a linear code C, then H = [−AT |In−k] is a parity check matrix for C.
Example 2.27. [22] The following matrices are respectively generator and parity check matrices for a [7,4,3] binary code.
G=
1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1
H=
0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1
.
The next result indicates the relationship between a linear code and a set of independent vectors, which we make use of in Section 6.4 to construct sets of frequency rectangles. First, we give here the definition of t–independent vectors.
Definition 2.19. A set S of vectors in (Fq)n is said to be t–independent if each of its subset of size t is linearly independent.
Theorem 2.28. [10] A linear code C with a parity check matrix H has min- imum distance d if and only if the matrix H contains a set of d dependent columns but each set of d−1 columns is independent.
In the theory of codes, for fixed parameters n and M, a code with the highest possibledis desirable. Similarly, for fixednandd, a code of maximum size is preferred. Therefore a significant effort has been put into finding the bounds in terms of these parameters. Here we include some of these results.
Theorem 2.29 (Singleton bound). [3] If C is an (n, M, d)q-ary code, then M ≤qn−d+1. In the case of linear [n, k, d] code, d≤n−k+ 1.
Theorem 2.30 (Sphere-packing bound). [3]If C is an(n, M, d)q-ary code of packing radius ρ=⌊(d−1)/2⌋, then
M
1 + (q−1)n+ (q−1)2 n
2
+· · ·+ (q−1)ρ n
ρ
≤qn,
and in the linear case:
ρ
X
i=0
(q−1)i n
i
≤qn−k.
The next result gives a lower bound on n for a linear [n, k, d] code, given its minimum weight and dimension.
Theorem 2.31 (Griesmer Bound). [22]Let C be an[n, k, d]q-ary code with k ≥1. Then
n≥
k−1
X
i=0
d qi
.
We refer the reader to [22] for a detailed overview of research in this field.
A useful online repository for the bounds on codes over finite fields of size less than or equal to 9 is available at [16].
References
[1] S. Addelman and O. Kempthorne. Some main-effect plans and orthogonal arrays of strength two. Annals of Mathematical Statistics, 32(4):1167–
1176, 1961.
[2] S. S. Agaian. Hadamard Matrices and Their Application. Springer, 1985.
[3] E. F. Assmus and J. D. Key. Designs and their Codes. Number 103.
Cambridge University Press, 1994.
[4] R. C. Bose and S. S. Shrikhande. On the falsity of Euler’s conjecture about the non-existence of two orthogonal latin squares of order 4t+ 2.
Proceedings of the National Academy of Sciences of the United States of America, 45(5):734, 1959.
[5] R. C. Bose, S. S. Shrikhande, and E. T. Parker. Further results on the construction of mutually orthogonal latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics, 12:189–203, 1960.
[6] T. Britz, N. J. Cavenagh, A. Mammoliti, and I. M. Wanless. Mutually orthogonal binary frequency squares. The electronic journal of combina- torics, 27(3), 2020.
[7] B. W. Brock. Hermitian congruence and the existence and completion of generalized hadamard matrices. Journal of Combinatorial Theory, Series A, 49(2):233–261, 1988.
[8] S. Cammann. The evolution of magic squares in China. Journal of the American Oriental Society, 80(2):116–124, 1960.
[9] C.-S. Cheng. Orthogonal arrays with variable numbers of symbols. The Annals of Statistics, 8:447–453, 1980.
[10] C. J. Colbourn and J. H. Dinitz. Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications). Chapman
& Hall/CRC, 2006.
[11] R. Craigen, G. Faucher, R. Low, and T. Wares. Circulant partial hadamard matrices. Linear Algebra and its Applications, 439(11):3307–
3317, 2013.
[12] R. Craigen, J. Seberry, and X.-M. Zhang. Product of four hadamard ma- trices. Journal of Combinatorial Theory, Series A, 59(2):318–320, 1992.
[13] D. Z. Djokovic. Hadamard matrices of order 764 exist. Combinatorica, 28:487–489, 2008.
[14] W. Federer, A. Hedayat, and J. Mandeli. Pairwise orthogonal f-rectangle designs.Journal of statistical planning and inference, 10(3):365–374, 1984.
[15] J. Goethals and J. J. Seidel. Orthogonal matrices with zero diagonal.
Canadian Journal of Mathematics, 19:1001–1010, 1967.
[16] M. Grassl. Code tables: Bounds on the parameters of various types of codes. Accessed Nov 2022. http://codetables.markus-grassl.de/.
[17] J. Hadamard. R´esolution d’une question relative aux d´eterminants. Bull.
des Sciences Math, 17:240–246, 1893.
[18] A. Hedayat, D. Raghavarao, and E. Seiden. Further contributions to the theory of f-squares design. The Annals of Statistics, 3:712–716, 1975.
[19] A. Hedayat and E. Seiden. f-square and orthogonal f-squares design: A generalization of latin square and orthogonal latin squares design. The Annals of Mathematical Statistics, 41(6):2035–2044, 1970.
[20] S. Hedayat, Sloane. Orthogonal Arrays. Springer, New York, NY, 1999.
[21] K. J. Horadam. Hadamard matrices and their applications. Princeton university press, 2012.
[22] W. Huffman, J. Kim, and P. Sole.Concise Encyclopedia of Coding Theory.
CRC Press, 2021.
[23] D. Jungnickel. On difference matrices, resolvable transversal designs and generalized hadamard matrices. Mathematische Zeitschrift, 167(1):49–60, 1979.
[24] A. D. Keedwell and J. D´enes. Latin squares and their applications. Else- vier, 2015.
[25] H. Kharaghani and B. Tayfeh-Rezaie. A hadamard matrix of order 428.
Journal of Combinatorial Designs, 13(6):435–440, 2005.
[26] C. Laywine. A geometric construction for sets of mutually orthogonal frequency squares. Utilitas Mathematica, 35:95–102, 1989.
[27] C. Laywine and G. Mullen. Discrete Mathematics Using Latin Squares.
1484 Series. Wiley, 1998.
[28] C. F. Laywine and G. L. Mullen. Generalizations of Bose’s equivalence between complete sets of mutually orthogonal latin squares and affine planes. Journal of Combinatorial Theory, Series A, 61(1):13–35, 1992.
[29] C. F. Laywine and G. L. Mullen. A table of lower bounds for the number of mutually orthogonal frequency squares. Ars Combinatoria, 59:85–96, 2001.
[30] M. Li, Y. Zhang, and B. Du. Some new results on mutually orthogonal frequency squares. Discrete Mathematics, 331:175–187, 2014.
[31] S. London. Constructing new Turyn type sequences, T-sequences and Hadamard matrices. PhD thesis, University of Illinois at Chicago, 2013.
[32] J. Mandeli. Complete-sets of mutually orthogonal frequency rectangle designs having twice a prime power number of columns. Utilitas Mathe- matica, 41:151–160, 1992.
[33] J. Mandeli and W. Federer. On the construction of mutually orthogonal f-hyperrectangles. Utilitas Math., 25:315––324, 1984.
[34] V. C. Mavron. Frequency squares and affine designs. The Electronic Journal of Combinatorics, 7(1):56, 2000.
[35] G. L. Mullen. Polynomial representation of complete sets of mutually orthogonal frequency squares of prime power order.Discrete Mathematics, 69(1):79–84, 1988.
[36] R. E. Paley. On orthogonal matrices.Journal of Mathematics and Physics, 12(1-4):311–320, 1933.
[37] C. R. Rao. Factorial experiments derivable from combinatorial arrange- ments of arrays.Supplement to the Journal of the Royal Statistical Society, 9(1):128–139, 1947.
[38] J. Williamson. Hadamard’s determinant theorem and the sum of four squares. Duke Mathematical Journal, 11(1):65–81, 1944.