Linear algebra and in particular vector spaces are important mathematical structures and have numerous applications. There are areas, for example coding theory, finite geometries and statistical theory of designs, where vector spaces provide basic foundations. There it is felt that higher order and more efficient structures can be developed with advantage by suitably composing the lower order structures. There is thus interest in developing new mathematical composition laws on vectors, matrices and polynomials. Elias (1954) used Kronecker product of matrices to develop higher efficiency codes by combination of lower order codes. The method was powerful, but it could develop only a sparse class of product codes and the all important duality property was lost. Sharma (2008), introduced a new concept of matrix-multiplication called the partitioned product of matrices, and obtained a rather large class of rank-partitioned product codes in which the duality property was preserved. Cyclic codes, as is known, are ideally suited for implementation through shift registers. Most important codes like BCH, Goppa codes etc. are cyclic codes, which are best characterized through their generator polynomials. Composing higher order cyclic codes from lower order codes in terms of their generator polynomials has not been explored at any length. The reason for not been able to do that is that there does not exist a method of multiplying two polynomials leading to what may be used to consider product of two cyclic codes. In this paper, we start by introducing a new product of two polynomials defined over a field. It is a generalization of the ordinary product. For convenience we call the two polynomials as outer and inner polynomials. The new defined product then results in non-overlapping segments obtained by multiplying it with coefficients of outer polynomials and expanding powers of the variable. It is called Ordered Power Product and has elegant algebraic properties leading to new algebraic structures. Paper carries a section on applications of the above concepts in developing product of two cyclic codes, in terms of the new product defined by us of two polynomials. Paper is organized as follows: Section 2 gives basic definitions required for later study. In Section 3, the Ordered Power Product of two polynomials in introduced and its algebraic properties are reported. Section 4 carries Applications of Ordered Power Product of polynomials in Coding Theory and an example is given to illustrate the a cyclic code arising from the OPP of two cyclic codes. In Section 5, we refer to further work under study.
2. Main results
For our purpose, we begin with following simple well known algebraic ideas of polynomials over a field F. For details, one may refer to any standard text on Algebra as also Peterson and Weldon 1972 Chapters 2 and 6.
Definition 2.1. Ring of polynomials: Given a field F, we can consider a polynomial as,
where coefficients a0, a1, a2, ..., an−1 ∈ F. The collection of all polynomials over field F, denoted by F [x] under usual addition and product of polynomials, forms a ring, called ‘ring of polynomials over F’.
Definition 2.2. Ring of polynomials : Algebra of Polynomial Residue Classes (Refer Peterson andWeldon (1972) The residue classes of the ring of polynomials F [x] modulo (xn − 1) a polynomial ƒ (x) of degree n form a commutative linear algebra of dimension n over the coefficient field F.
Definition 2.3. Cyclic code: An(n, k) block code C is said to be cyclic if it is linear and if for every code word C = (c0, c1, c2, ..., cn−1) ∈ C, its right cyclic shift = (cn−1, c1, c2, ..., cn−2) ∈ . An cyclic code is generated by a polynomial g (x) of degree n − k that is a factor of (xn − 1).
3. New Product - Ordered Power Product of polynomials
Linear algebraic structures, as is well known, are studied with advantage through polynomials. While direct polynomial addition and multiplications are commonly used as the operations on polynomial, wider algebraic structures and applications are done by considering polynomial algebra over a field. Here we shall consider polynomials over a field F, and consider the set of all polynomials F [x]. We will be considering structure of F [x] modulo some polynomial, say, g (x). It is proposed to define a new type of composition on polynomials, in which order of the polynomials multiplied is retained and segments of the product arise in terms of rising powers, we name it as ‘Ordered Power product.’ To motivate the new product, let us look at ordinary product of two polynomials λ (x) = c0+c1x+c2x2+...+cm−1xm−1 and V (x) = a0+a1x+a2x2+...+an−1xn−1, then
Generalizing this by considering suitably indexed powers of x in the segments above, we arrive at the following what we call as the ‘Ordered power Product’ (OPP).
Definition 3.1. Ordered Power Product of Polynomials in non-overlapping sifting segments: Let λ (x) = c0+c1x+c2x2+...+cm−1xm−1 and V (x) = a0+a1x+a2x2+...+an−1xn−1 be two polynomials of degree m− 1 and n − 1 respectively over a field F. The ordered power product of λ (x)and V (x) (in nth power of in different segments), denoted by λ (x) ∗ V (x), is defined as
In short we shall call it ‘Ordered Power Product’ (OPP) or ∗product of λ (x) and V (x).
Remark 3.1. Clearly, the degree of the product polynomial λ (x) ∗ V (x) is (m − 1) n + (n − 1) = mn − 1 .
Remark 3.2. We call it ‘ordered’ because as proved below, unlike ordinary product it is not commutative.
Remark 3.3. Just for convenience, in the product so defined, we will refer to λ (x) as outer polynomial and V (x) as inner polynomial.
Example 1. Let λ (x) = 1 + 2x + 4x3 + 5x4be a polynomial (outer) of degree 4, with m = 5 and V (x) = 2+x2 +x6be the polynomial of degree 6 with n = 7 then
this example also verifies the degree of outer product of two polynomials λ (x) ∗ V (x) mentioned above.
Definition 3.2. Vector Equivalence of OPP: Interestingly, if polynomials are replaced by their equivalent vectors
then it can be easily seen that λ ∗ V is the Kronecker product of vectors λ and V .
Properties of Ordered Power of Product of Polynomials
(i) Non Commutativity: The * product of polynomials is in general non commutative, i.e.,
Proof: Choosing λ (x) and V (x) as above, we have
In general (1) and (2) are different and this proves the result.
(ii) Associativity : The * product of two polynomials is in general associative, i.e.,
Proof: Associativity: in the above notation associativity is easy to prove by calculations like
(iii) Distributive over addition: The * product of two polynomials is such that:
(a) The outer polynomial distributes over the sum of inner polynomials,
(b) The inner polynomial distributing over the sum of outer polynomials,
Where U (x) and V (x) are of same degree n−1 and C (x) and D(x) are of same degree m − 1 .
Proof: We prove both the forms
4. Ordered Power Product of Rings of Polynomials
In this section we extend the idea considered above to product of two sets of polynomials defined over the same field F. In particular, let us consider two rings of polynomials namely, Rm [x] modulo (xm − 1) and Rn [x] modulo (xn − 1), where m and n are any positive integers. Obviously these rings contain respectively all polynomials of degree m − 1, n − 1 and less.
Definition 4.1. Let λj (x) = c0j +c1jx+c2jx2+...+c(m−1)jxm−1 ∈ Rm (x) for different values of j be polynomials of degree m − 1 or less and Vi (x) = a0i+a1ix+a2ix2+...+a(n−1) ixn−1 ∈ Rn (x) for different values ofi, be polynomials of degree n − 1 or less. We define the set of ordered power product of Rm [x] and Rn [x] as
Theorem 4.2. If Rm [x] F (x) modulo (xm - 1), Rn [x] = F (x) modulo (xn - 1), m and n being any positive integers, are two rings of polynomial over a field F, then
is a ring of polynomials of degree mn - 1, under ordinary addition and ordered product of elements of Γ (x) modulo (xm − 1, xn − 1) over F.
Proof. Proof is simply followed by the definition.
5. Applications of OPP of Polynomials in Coding Theory
In this section, we consider cyclic binary codes represented by their generator polynomials and study the codes obtained by their OPP-composition. It may be pointed out that in the binary case, (xn − 1) may be taken as (xn + 1) , and that 1+1 = 0. Since cyclic codes of length n are generated by factors of (xn − 1), we next give some useful results for getting its factors.
Lemma 5.1. For p and q distinct prime numbers and r and s are natural num-bers, we have
Proof. The results can be easily verified, and extended for , for any n when n is expressed as product of powers of primes.
Theorem 5.2. If (n1, k1) and ((n2, k2) are two cyclic codes with g1 (x) and g2 (x) as their generator polynomials then the * product of their generator polynomials given by g (x) = g1 (x)∗g2 (x) of degree (n1 − k1) n2+(n2 − k2)code, will generate a cyclic(n, k) code, where n = n1n2 and k = (k1 − 1) n2 + k2, whenever g2 (x) divides xn − 1.
Proof. Let g (x) = g1 (x) ∗ g2 (x). It can be written as a simple product: g (x) = g1 (xn2) g2 (x) where g1 (x) and g2 (x) are divisors of xn1−1 and xn2−1 respectively. For g (x)to represent a cyclic code of lengthn1n2−1, g (x) must a factor of xn1n2−1. So we examine the conditions under which g (x) = g1 (x) ∗ g2 (x) is a divisor of xn1n2−1. First we show that g1 (xn2) is a divisor of xn1n2−1. This follows from the fact that g1 (x) is a divisor of xn1−1, and by replacing x by xn2 . Thus we get that g1 (xn2) is a divisor of (xn2)n1−1 = xn1n2 − 1. It remains to investigate the conditions forg2 (x) is a factor of xn1n2 − 1. This will not happen for all choices of n1n2, as is evident by the Example 3 below. We therefore can use the result of the Lemma above for determining ifg2 (x) divides xn1n2 − 1. In fact any one or products of any numbers of these factors can be taken forg2 (x) to get the *-product cyclic code from generator polynomials of component codes.
Note: It may have been noted that while choice of , the inner-polynomial, is limited, there is no limitation on choice of outer polynomial in forming OPP codes.
Theorem 5.3. If (n1, k1)and(n2, k2) are two cyclic codes with g1 (x)andg2 (x) as their generator polynomials and g2 (x)dividesxn1n2 − 1 then the * product g (x) = g1 (x) * g2 (x) generates (n1n2, k)code, where k = (k1 − 1) n2 + k2. Also then k − 1 code polynomials g (x), xg (x), x2g (x), ..., xk−1g (x) span cyclic code C.
Proof. The proof follows directly from the definition of cyclic code generated by g(x).
Example 2. Let us consider two cyclic codes C1andC2 where C1 is (7, 4) code and C2 is (3, 1) cyclic code with generator polynomials g1 = (1 + x + x3) and g2 = (1 + x + x2) respectively. Then
Example 3. Let us consider two cyclic codes C1andC2 where C1 is (3, 2) code and C2 is ( (7, 4) cyclic code with generator polynomials g1 = (1 + x) and g2 = (1 + x + x3) respectively. Then
Here g2 (x) = (1 + x + x3) is not a divisor of x21 − 1 .
Theorem 5.4. If two linear binary cyclic codes (n1, k1) and (n2, k2) have rates R1 and R2 then the rate R of their cyclic Code is an increasing function of the rate of either component code.
Proof. Let R be the rate of the product cyclic code. Then, we have
and trivially, (R = R1R2), when k = 1. In general the difference is directly proportional to R1, and increases with R2. This shows that the newly introduced OPP cyclic codes are a new class of codes that have rates better than those that be obtained by their Kronecker product codes.
6. Concluding Remarks
We have considered ordered power product of two polynomials in such powers of x that the various segments of the inner polynomial are laterally advanced without having overlaps amongst them, with coefficients multiplied by those of the outer polynomial. For this choice the motivation was to consider a new composition suitable for developing new product type of codes, which are more efficient. However, an ordered power product may be defined more generally, when this is not the case, that is when λ (x) = c0+c1x+c2x2 +...+cm−1xm−1 and V (x) = a0 + a1x + a2x2 + ... + an−1an−1; but λ (x) ∗ V (x) = c0V (x) + c1xkV (x) + c2x2kV (x) + ... + cm−1x(m−1)kV (x). where k < n, or even when k > n . In such cases we can perhaps use the notation for OPP using an index like ∗k −opp. These interesting cases are being studied separately.