Let R be a Galois ring of characteristic $$p^a$$, where p is a prime and a is a natural number. I... more Let R be a Galois ring of characteristic $$p^a$$, where p is a prime and a is a natural number. In this paper cyclic codes of arbitrary length n over R have been studied. The generators for such codes in terms of minimal degree polynomials of certain subsets of codes have been obtained. We prove that a cyclic code of arbitrary length n over R is generated by at most $$min\{a, t+1\}$$ elements, where $$t=max\{deggx\},$$gx a generator. In particular, it follows that a cyclic code of arbitrary length n over finite fields is generated by a single element. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Grobner basis.
We study the structure of cyclic codes of length k 2 over 8 Z for any natural number k. It is kno... more We study the structure of cyclic codes of length k 2 over 8 Z for any natural number k. It is known that cyclic codes of length k 2 over 8 Z are ideals of the ring R= ! 1 2 k cyclic codes of length k 2 over 8 Z are generated as ideals by at most three elements.
Let R n be the n-dimensional Euclidean space. Let L denote a lattice in R n of determinant 1 such... more Let R n be the n-dimensional Euclidean space. Let L denote a lattice in R n of determinant 1 such that there is a sphere centered at the origin O which contains n linearly independent points of L on its boundary but no point of L other than O inside it. A wellknown conjecture in the geometry of numbers asserts that any closed sphere in R n of radius 1 2 √ n contains a point of L. This is known to be true for n 6. Here we give a unified simple proof for n 6 of the more general conjecture of Woods.
Recent studies involve various approaches to establish a generating set for cyclic codes of arbit... more Recent studies involve various approaches to establish a generating set for cyclic codes of arbitrary length over the class of Galois rings. One such approach involves the use of polynomials with minimal degree corresponding to specific subsets of the code, defined progressively. In this paper, we extend this approach to obtain a set of generators of cyclic codes over finite chain rings. Further, we observe that this set acts as a minimal strong Gröbner basis(MSGB) for the code.
Determining Minimal Degree Polynomials of a Cyclic Code of Length 2^k over \mathbb Z_8
The rank of a cyclic code of length \(n=2^k\) over \({\mathbb {Z}}_8\) is \(n-v\) where v is the ... more The rank of a cyclic code of length \(n=2^k\) over \({\mathbb {Z}}_8\) is \(n-v\) where v is the degree of a minimal degree polynomial in the code. In this paper, minimal degree polynomials in a cyclic code C of length \(n = 2^k\) (where k is a natural number) over \({\mathbb {Z}}_8\) are determined. Further, using these minimal degree polynomials, all 95 (46 principally generated and 49 non principally generated) cyclic codes of length 4 over \({\mathbb {Z}}_8\) are calculated in terms of their distinguished sets of generators.
In this paper, the structure of cyclic codes over m 2 of length k 2 for any natural number k is s... more In this paper, the structure of cyclic codes over m 2 of length k 2 for any natural number k is studied. It is proved that cyclic codes over 1 2 n x x Z R m / ] [ of length n = k 2 are generated by at most m elements. Keywords— Codes, Cyclic codes, Ideals, Principal ideal Ring
In this paper, a set of generators (in a unique from) called the distinguished set of generators,... more In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length \(n = 2^k\) (where k is a natural number) over \(Z_8\) is obtained. This set of generators is used to find the rank of the cyclic code C. It is proved that the rank of a cyclic code C of length \(n=2^k\) over \(Z_8\) is equal to \(n-v\), where v is the degree of a minimal degree polynomial in C. Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length \(n=2^k\) over \(Z_8\) is given. An example of the best codes over \(Z_8\) of length 4 having largest minimum Hamming, Lee and Euclidean distances among all codes of the same rank is also given.
Linear complementary dual(LCD) codes are being utilized to protect cryptographic implementations ... more Linear complementary dual(LCD) codes are being utilized to protect cryptographic implementations from side-channel and fault attacks. In this work, LCD codes over the ring Z 4 are examined. It is observed that an LCD cyclic code over Z 4 of odd length is always reversible. However, the converse of this result does not hold. A suitable example to validate this statement is presented. Further, a necessary and sufficient condition for a reversible cyclic code of odd length to be an LCD code is obtained. Finally, it is found that among cyclic codes of length 2 k , no non-trivial LCD code exists.
It is observed that density plots of any mathematical function, being an arrangement of shades or... more It is observed that density plots of any mathematical function, being an arrangement of shades or colours in rectangular meshes is an image which can actually be implemented on different types of looms. Further, it is noted that using some simple techniques and simple properties of Mathematics, beautiful and innovative graphic designs can be created. A few such designs are actually created by using the density plot command from Mathematica. These designs are innovative in the sense that instead of conventional 4 -fold, 6-fold or 8-fold symmetries, these designs display 3-fold, 5-fold or 7-fold symmetries. I. INTRODUCTION Any design or pattern that is given as an arrangement of colours or shades in the form of a rectangular mesh can easily be transferred and implemented on many types of looms and can be readily used in different types of Textile Industries. It is noted that density plot of a mathematical function is an arrangement of shades (default- greylevel ) in an rectangular arr...
Let R be a Galois ring of characteristic $$p^a$$, where p is a prime and a is a natural number. I... more Let R be a Galois ring of characteristic $$p^a$$, where p is a prime and a is a natural number. In this paper cyclic codes of arbitrary length n over R have been studied. The generators for such codes in terms of minimal degree polynomials of certain subsets of codes have been obtained. We prove that a cyclic code of arbitrary length n over R is generated by at most $$min\{a, t+1\}$$ elements, where $$t=max\{deggx\},$$gx a generator. In particular, it follows that a cyclic code of arbitrary length n over finite fields is generated by a single element. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Grobner basis.
We study the structure of cyclic codes of length k 2 over 8 Z for any natural number k. It is kno... more We study the structure of cyclic codes of length k 2 over 8 Z for any natural number k. It is known that cyclic codes of length k 2 over 8 Z are ideals of the ring R= ! 1 2 k cyclic codes of length k 2 over 8 Z are generated as ideals by at most three elements.
Let R n be the n-dimensional Euclidean space. Let L denote a lattice in R n of determinant 1 such... more Let R n be the n-dimensional Euclidean space. Let L denote a lattice in R n of determinant 1 such that there is a sphere centered at the origin O which contains n linearly independent points of L on its boundary but no point of L other than O inside it. A wellknown conjecture in the geometry of numbers asserts that any closed sphere in R n of radius 1 2 √ n contains a point of L. This is known to be true for n 6. Here we give a unified simple proof for n 6 of the more general conjecture of Woods.
Recent studies involve various approaches to establish a generating set for cyclic codes of arbit... more Recent studies involve various approaches to establish a generating set for cyclic codes of arbitrary length over the class of Galois rings. One such approach involves the use of polynomials with minimal degree corresponding to specific subsets of the code, defined progressively. In this paper, we extend this approach to obtain a set of generators of cyclic codes over finite chain rings. Further, we observe that this set acts as a minimal strong Gröbner basis(MSGB) for the code.
Determining Minimal Degree Polynomials of a Cyclic Code of Length 2^k over \mathbb Z_8
The rank of a cyclic code of length \(n=2^k\) over \({\mathbb {Z}}_8\) is \(n-v\) where v is the ... more The rank of a cyclic code of length \(n=2^k\) over \({\mathbb {Z}}_8\) is \(n-v\) where v is the degree of a minimal degree polynomial in the code. In this paper, minimal degree polynomials in a cyclic code C of length \(n = 2^k\) (where k is a natural number) over \({\mathbb {Z}}_8\) are determined. Further, using these minimal degree polynomials, all 95 (46 principally generated and 49 non principally generated) cyclic codes of length 4 over \({\mathbb {Z}}_8\) are calculated in terms of their distinguished sets of generators.
In this paper, the structure of cyclic codes over m 2 of length k 2 for any natural number k is s... more In this paper, the structure of cyclic codes over m 2 of length k 2 for any natural number k is studied. It is proved that cyclic codes over 1 2 n x x Z R m / ] [ of length n = k 2 are generated by at most m elements. Keywords— Codes, Cyclic codes, Ideals, Principal ideal Ring
In this paper, a set of generators (in a unique from) called the distinguished set of generators,... more In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length \(n = 2^k\) (where k is a natural number) over \(Z_8\) is obtained. This set of generators is used to find the rank of the cyclic code C. It is proved that the rank of a cyclic code C of length \(n=2^k\) over \(Z_8\) is equal to \(n-v\), where v is the degree of a minimal degree polynomial in C. Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length \(n=2^k\) over \(Z_8\) is given. An example of the best codes over \(Z_8\) of length 4 having largest minimum Hamming, Lee and Euclidean distances among all codes of the same rank is also given.
Linear complementary dual(LCD) codes are being utilized to protect cryptographic implementations ... more Linear complementary dual(LCD) codes are being utilized to protect cryptographic implementations from side-channel and fault attacks. In this work, LCD codes over the ring Z 4 are examined. It is observed that an LCD cyclic code over Z 4 of odd length is always reversible. However, the converse of this result does not hold. A suitable example to validate this statement is presented. Further, a necessary and sufficient condition for a reversible cyclic code of odd length to be an LCD code is obtained. Finally, it is found that among cyclic codes of length 2 k , no non-trivial LCD code exists.
It is observed that density plots of any mathematical function, being an arrangement of shades or... more It is observed that density plots of any mathematical function, being an arrangement of shades or colours in rectangular meshes is an image which can actually be implemented on different types of looms. Further, it is noted that using some simple techniques and simple properties of Mathematics, beautiful and innovative graphic designs can be created. A few such designs are actually created by using the density plot command from Mathematica. These designs are innovative in the sense that instead of conventional 4 -fold, 6-fold or 8-fold symmetries, these designs display 3-fold, 5-fold or 7-fold symmetries. I. INTRODUCTION Any design or pattern that is given as an arrangement of colours or shades in the form of a rectangular mesh can easily be transferred and implemented on many types of looms and can be readily used in different types of Textile Industries. It is noted that density plot of a mathematical function is an arrangement of shades (default- greylevel ) in an rectangular arr...
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