Rank Distance Bicodes and Their Generalization by W.B. Vasantha Kandasamy, Florentin Smarandache, N. Suresh

By W.B. Vasantha Kandasamy, Florentin Smarandache, N. Suresh Babu, R.S. Selvaraj

The hot type of rank distance m-codes may be helpful within the m-public key m-cryptosystems (m>1) and m-redundant array of cheap disks. AMRD-m-codes can be utilized in information multi (m)-storage platforms. those m-codes supply a cost-effective, time-saving substitute.

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By W.B. Vasantha Kandasamy, Florentin Smarandache, N. Suresh Babu, R.S. Selvaraj

The hot type of rank distance m-codes may be helpful within the m-public key m-cryptosystems (m>1) and m-redundant array of cheap disks. AMRD-m-codes can be utilized in information multi (m)-storage platforms. those m-codes supply a cost-effective, time-saving substitute.

Show description

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Extra resources for Rank Distance Bicodes and Their Generalization

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These bicodes will be useful in application where one set of code never attains the singleton-style bound were as the other code attains the singleton style bound. The special nature of these codes will be very useful in applications were two types of RD codes are used simultaneously. Next we proceed on to define the notion of semi circulant rank bicode of type I and type II. 20: Let C1 = C1[n1, k1] be a RD-code and C2 be a circulant rank code both are subspaces of rank spaces defined over the same field GF(2N).

The characterization is based on the condition for the minimum distance proved by Gabidulin in [24, 27]. 4: Let H = H1 ‰ H2 = (D ij1 ) ‰ (D ij2 ) be a 3 u n1 ‰ 3 u n2 bimatrix of birank 3 over GF(2N); n1 d N and n2 d N which satisfies the following condition. For any two distinct, non empty bisubsets P1, P2 where P1 P11 ‰ P21 and P2 P12 ‰ P22 of {1, 2, …, n1} and {1, 2, 3, …, n2} respectively there exists i1 i11 ‰ i21 , i2 i12 ‰ i22  {1, 2, 3} ‰ {1, 2, 3} such that, § · § · 1 1 2 2 ¨¨ ¦ D 1 1 ˜ ¦ D 1 1 ¸¸ ‰ ¨¨ ¦ D 2 2 ˜ ¦ D 2 2 ¸¸ z © j11P11 i1 j1 k11P21 i2 k1 ¹ © j12 P12 i1 j1 k22 P12 i2 k2 ¹ § · § · 1 1 2 2 ¨¨ ¦ D 1 1 ˜ ¦ D 1 1 ¸¸ ‰ ¨¨ ¦ D 2 2 ˜ ¦ D 2 2 ¸¸ .

We analogous to the definition given in MacWilliams and Sloane [61] define circulant bimatrix associated with a bivector in GF(2N) ‰ GF(2N). 14: A bimatrix of the from ª a0 «a « N 1 « # « ¬ a1 a1 ! aN 1 º ª b0 b1 ! bN 1 º a0 ! aN  2 »» ««bN 1 b0 ! bN  2 »» ‰ # # » « # # # » » « » a2 ! a0 ¼ ¬ b1 b2 ! b0 ¼ is called the circulant bimatrix associated with the bivector (a0, a1, …, aN–1) ‰ (b0, b1, …, bN–1)  GF(2N) ‰ GF(2N). Thus with each D = D1 ‰ E2  GF(2N) ‰ GF(2N), we can associate a T1 T2 circulant bimatrix whose ith bicolumn represents D i C1 ‰ E i C2 ; i = 0, 1, 2, …, N – 1.

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