### Download Super Special Codes Using Super Matrices by W. V. Vasantha Kandasamy, Florentin Smarandache, K. PDF

• March 28, 2017
• Nonfiction 2
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By W. V. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral

Tremendous specific codes are developed for the 1st time utilizing super-matrices. a number of fascinating homes of those codes, in addition to their purposes to different fields are mentioned during this publication.

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Additional info for Super Special Codes Using Super Matrices

Example text

We see that Vn ≅ Fqn as both are vector spaces defined over the same field Fq. Let Γ be an isomorphism Γ(v0, v1, …, vn–1) → {v0 + v1x + v2x2 + … + vn–1xn–1}. , w (v0, v1, …, vn–1) = v0 + v1x + … + vn–1x n–1. Now we proceed onto define the notion of a cyclic code. 7: A k-dimensional subspace C of Fqn is called a cyclic code if Z(v) ∈ C for all v ∈ C that is v = v0, v1, …, vn–1 ∈ C implies (vn–1, v0, …, vn–2) ∈ C for v ∈ Fqn . We just give an example of a cyclic code. 8: Let C ⊆ F27 be defined by the generator matrix ⎡1 1 1 0 1 0 0 ⎤ ⎡ g (1) ⎤ ⎢ ⎥ G = ⎢0 1 1 1 0 1 0 ⎥ = ⎢ g (2) ⎥ .

X + y, z〉p = 〈x, z〉p + 〈y, z〉p for all x, y, z ∈ V. 4. 〈x, y + z〉p = 〈x, y〉p + 〈x, z〉p for all x, y, z ∈ V. 5. x, y〉p = α 〈x, y〉p and 6. y〉p = β〈x, y〉p for all x, y, ∈ V and α, β ∈ Fp . Let V be a vector space over a field Fp of characteristic p, p is a prime; then V is said to be a pseudo inner product space if there is a pseudo inner product 〈,〉p defined on V. We denote the pseudo inner product space by (V, 〈,〉p). Now using this pseudo inner product space (V, 〈,〉p) we proceed on to define pseudo-best approximation.

We know that Fqn is a finite dimensional vector space over Fq. If we take Z2 = (0, 1) the finite field of characteristic two. Z52 = Z2 × Z2 × Z2 × Z2 × Z2 is a 5 dimensional vector space over Z2. Infact {(1 0 0 0 0), (0 1 0 0 0), (0 0 1 0 0), (0 0 0 1 0), (0 0 0 0 1)} is a basis of Z52 . Z52 has only 25 = 32 elements in it. Let F be a field of real numbers and V a vector space over F. An inner product on V is a function which assigns to each ordered pair of vectors α, β in V a scalar 〈α /β 〉 in F in such a way that for all α, β, γ in V and for all scalars c in F.