Special Classes of Set Codes and Their Applications

In this book the authors introduce the notion of set codes, set bicodes and set n-codes. These are the most generalized notions of semigroup n-codes and group n-codes. Several types of set ncodes are defined. Several examples are given to enable the reader to understand the concept. These new classes of codes will find applications in cryptography, computer networking (where fragmenting of codes is to be carried out) and data storage (where confidentiality is to be maintained). We also describe the error detection and error correction of these codes. The authors feel that these codes would be appropriate to the computer-dominated world. This book has three chapters. Chapter One gives basic concepts to make the book a self-contained one. In Chapter Two, the notion of set codes is introduced. The set bicodes and their generalization to set n-codes (n ≥ 3) is carried out in Chapter Three. This chapter also gives the applications of these codes in the fields mentioned above. Illustrations of how these codes are applied are also given. The authors deeply acknowledge the unflinching support of Dr.K.Kandasamy, Meena and Kama. This book is dedicated to the memory of the first author's father Mr.W.Balasubramanian, on his 100 th birth anniversary. A prominent educationalist, who also severed in Ethiopia, he has been a tremendous influence in her life and the primary reason why she chose a career in mathematics.

This chapter has two sections. In section one we introduce basic concepts about set vector spaces, semigroup vector space, group vector spaces and set n-vector space. In section two we recall the basic definition and properties about linear codes and other special linear codes like Hamming codes, parity check codes, repetition codes etc.

In this section we just recall the definition of linear algebra and enumerate some of its basic properties. We expect the reader to be well versed with the concepts of groups, rings, fields and matrices. For these concepts will not be recalled in this section.

Throughout this section, V will denote the vector space over F where F is any field of characteristic zero. DEFINITION 1.1.1: A vector space or a linear space consists of the following: i.

a field F of scalars. ii.

a set V of objects called vectors. iii.

a rule (or operation) called vector addition; which associates with each pair of vectors α, β ∈ V; α + β in V, called the sum of α and β in such a way that a. addition is commutative α + β = β + α. b. addition is associative α + (β + γ) = (α + β) + γ.

c. there is a unique vector 0 in V, called the zero vector, such that α + 0 = α for all α in V. d. for each vector α in V there is a unique vector -α in V such that α + (-α) = 0. e. a rule (or operation), called scalar multiplication, which associates with each scalar c in F and a vector α in V, a vector c α in V, called the product of c and α, in such a way that 1. 1 α = α for every α in V. It is important to note as the definition states that a vector space is a composite object consisting of a field, a set of ‘vectors’ and two operations with certain special properties. The same set of vectors may be part of a number of distinct vectors. We simply by default of notation just say V a vector space over the field F and call elements of V as vectors only as matter of convenience for the vectors in V may not bear much resemblance to any pre-assigned concept of vector, which the reader has.

Example 1.1.1: Let R be the field of reals. R[x] the ring of polynomials. R[x] is a vector space over R. R[x] is also a vector space over the field of rationals Q.

Example 1.1.2: Let Q[x] be the ring of polynomials over the rational field Q. Q[x] is a vector space over Q, but Q[x] is clearly not a vector space over the field of reals R or the complex field C.

Example 1.1.3: Consider the set V = R × R × R. V is a vector space over R. V is also a vector space over Q but V is not a vector space over C. Example 1.1.4: Let M m × n = {(a ij ) ⏐ a ij ∈ Q} be the collection of all m × n matrices with entries from Q. M m × n is a vector space over Q but M m × n is not a vector space over R or C.

. P 3 × 3 is a vector space over Q.

Example 1.1.6: Let Q be the field of rationals and G any group. The group ring, QG is a vector space over Q.

Remark: All group rings KG of any group G over any field K are vector spaces over the field K.

We just recall the notions of linear combination of vectors in a vector space V over a field F. A vector β in V is said to be a linear combination of vectors ν 1, …,ν n in V provided there exists scalars c 1 ,…, c n in F such that

. Now we proceed on to recall the definition of subspace of a vector space and illustrate it with examples. DEFINITION 1.1.2: Let V be a vector space over the field F. A subspace of V is a subset W of V which is itself a vector space over F with the operations of vector addition and scalar multiplication on V. DEFINITION 1.1.3: Let S be a set. V another set. We say V is a set vecto

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