Codes have been used with great efficiency in some important applications. Perhaps the most immediate application of codes is that which relates them with the encoding and decoding of ”messages”, and this will be the subject of the this section.
Of primary interest is the capability of a code that is constructed from a design.
We will assume that asymmetric q-ary channelis used, where each symbol in the alphabet of the code has the same probability of being transmitted erroneously and has the same probability to occur when an error has been made. We will describe the techniques ofsyndrome decodingandmajority logic decoding. These methods are often used in decoding projective geometry codes.
CHAPTER 4. LINKS OF CODES AND OTHER COMBINATORIAL STRUCTURES49
The following result establishes the exact measurement of the error-detection and error-correction capability of a code assuming the use of a symmetric channel.
The proof can be found in any standard text in coding theory: see for example [4, Chapter 2].
Theorem 4.28. Let C be a code with minimum distanced. ThenC can detect d−1 errors and correct bd−12 c errors.
Proof: See [4].
4.6.1 Nearest neighbour decoding
The decoding scheme in which a received word y is decoded as the closest word in the q-ary code to y, should such a word be uniquely determined, is called nearest neighbour decoding. Here “close” is measured in terms of the Hamming distance between two codewords. Thus, the greater the minimum distance of a code, the larger the number of errors can be corrected. Assuming the use of the symmetric q- ary channel, this decoding algorithm maximizes the probability that, after decoding, the correct word is finally received. Note that for large codes this algorithm is costly as it requires a comparison between the received vector y and every codeword in the code. For a linear code, the syndrome of the received vector y, denoted Syn(y), can be used to reduce the number of comparisons that are needed and to reduce the amount of memory needed to implement nearest neighbour decoding. This method is referred to as syndrome decoding.
4.6.2 Majority logic decoding
The majority decoding schemes are useful in decoding several families of codes: see ([107, 47, 45]). We describe the one-step majority logic decoding algorithm. In [33, Section 3.3], Clark shows that this algorithm is an effective decoding scheme for binary codes of the projective planes. Multi-step majority decoding can be implemented with codes of designs from geometries: see [107].
CHAPTER 4. LINKS OF CODES AND OTHER COMBINATORIAL STRUCTURES50
Definition 4.29. A set of 1×n vectors {v1, v2, . . . , vr} is said to be orthogonal at position i if the vectors form an r×n matrix with all entries in the ith column equal to 1, and every column has either all zeros or exactly one 1 and r−1 zeros.
Let x be the sent codeword of length n, y the received vector, and suppose that there are at most t errors. Then x+e = y where e has non-zero entries at the coordinate positions where the errors have occurred. Also, y·v = (x+e)·v = x·v+e·v =e·v for every vectorv ∈C⊥. Suppose there areri vectors{v1, v2, . . . , vri} in the dual code C⊥ of C that are orthogonal at positioni, where 1≤i≤n.
If an error occurred at theithposition, then there are at leastri−(t−1) equations (check equations) of the systems Si = {y·vj|j ∈ {1,2, . . . , ri}} whose value is ei. In order to correct the errors that have occurred, we must have a clear majority of the check equations in Si that equal ei. Thus we require,t−1< ri−(t−1) so that ri >2(t−1).
If no error occurred at theith position, then there are at most t check equations in Si that will be non-zero for 1 ≤ i ≤ n. This means that at least ri −t check equations will be 0 for each i. For a clear majority of the checks to be 0 we need t ≤ri −t. Hence ri ≥2t.
It follows that if there are at mostt ≤ r2i errors introduced and there arerivectors in the dual code C⊥ of C that are orthogonal at position i,then the majority logic decoding algorithm can detect and correct an error made in the ith position. If such a set of checks exist for every position i ∈ {1,2,· · · , n}, then we can correct up to t errors, where 2t ≤ r and r = Mini{ri}. If the minimum weight of C is d, then C can correct at most bd−12 c errors. So majority logic will use this capability as long as r≥ bd−12 c, that is, r≥ d−12 if d is odd and r≥ d−22 if d is even.
Chapter 5
Codes related to combinatorial structures
As a mathematical theory, coding theory is relatively young, with its roots in Shannon’s [109] seminal paper in 1948. The practical gains, due to coding, demonstrated there, and elsewhere since, have provided motivation for much of coding theory. It is fascinating how a large mathematical theory was and is continuing to be developed. The mathematical areas needed in classical coding have been mainly algebraic. However through time, subsequent developments have expanded this mathematical theory considerably. A frequent question in coding theory is
“how one constructs a code, or structure related to a code, that is optimal in some mathematical or applied sense.” In this chapter we consider some ways in which codes have been constructed.
5.1 Codes from combinatorial designs
Coding theory has made many contributions to the theory of combinatorial designs.
A code generated by the incidence matrix of designs has been useful in either constructing new designs or showing that certain designs do not exist, as it is for example the case of the projective plane of order 10. Coding theory has also been
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CHAPTER 5. CODES RELATED TO COMBINATORIAL STRUCTURES 52 used to extend designs. In [76] Kennedy and Pless extended designs ”held” by vectors of a code. The connection between designs and codes leads to the construction of new designs. Using the knowledge about codes and the existence of designs in codes can be useful for decoding purposes. For example a binary vector x of weight w is said to determine the block of w points corresponding to the positions where x has non-zero coordinates. In such case we say that vectors of a fixed weight w in a binary code of length n hold a t-design if the blocks determined by these vectors are the blocks of a t-design on n points. This means that there must exist t and λ so that every set of t coordinate positions occurs as non-zero positions for exactly λ vectors of weight w. The knowledge of the number of vectors of each weight existing in a code is crucial in determining whether or not the supports of these vectors could form a design. For q = 2 the supports are in a one-to-one correspondence with the codewords. The celebrated Assmus-Mattson Theorem ([4, Theorem 2.11.2]) establishes the connection between designs and codes, in that vectors of certain weight in a q-ary code hold a design, and we can determine the number of vectors of such weight. Notice that ifS is the support of a vector in a code overFq then it is the support of at least q−1 such vectors; in fact, preciselyq−1 vectors if the minimum weight of the code is |S|. For q= 2 the supports are in a one-to-one correspondence with the codewords. Once again the Assmus-Mattson Theorem gives conditions on the weight enumerators of a code and its dual that are sufficient to ensure that the support of the minimum weight vectors (and other weights also) yield a t-design where t is a positive integer less than the minimum weight.
For a general incidence structure D = (P,B,I) and any field F, we denote the vector space of functions from P to F by FP. For w ∈ FP, the value of w at the point p isw(p) in F.
Definition 5.1. Thesupport set of a function win FP is defined to be the subset of points inP whose images underware non-zero, that is, Supp(w)={p∈ P |w(p)6=
0}. The characteristic function for a block B is denoted by vB and defined to be
vB(p) =
1, if p∈B 0, if p /∈B.
CHAPTER 5. CODES RELATED TO COMBINATORIAL STRUCTURES 53
The standard basis for this vector space is {v{p}|p∈ P}.
Definition 5.2. A q-ary code of a design D = (P,B,I) is the subspace of the function space FqP generated by the characteristic functions of the blocks of D and is denoted by Cq(D).
If the point set of D is denoted by P and the block set by B, and if Q is any subset of P, then we will denote the incidence vector of Q by vQ. Thus CF(D) =
vB|B ∈ B
, and is a subspace of FP. The dimension of the codeCp(D) of the design D over a prime field Fp is the rank of the generating matrix of the code and is referred to as the p-rank of D.
In general the minimum weight is less than the block size of D, but for the p- ary codes of geometry designs, where p is the characteristic of the underlying field of the geometry, we have equality by the work of Delsarte et all: see [48] and [33, Section 6.1].
The following Lemma be found in [78], is an important result on the automorphism group of codes obtained from incidence structures.
Lemma 5.3. [78] Let C be the linear code of length n of an incidence structure I over a field F.Then the automorphism group of C is the full symmetric group if and only if C =Fn or C =h1i⊥.
Proof: See [78].