Constructing SCN bases in characteristic 2

A simple deterministic algorithm to construct a normal basis of GF(qⁿ) over GF(q) (q=p^r, p prime) is given. When p=2, the authors deduce a (self complementary normal-SCN) basis of GF(q ⁿ) over GF(q) for n odd, or n=2t, t odd. In characteristic 2 these cases are known to be the only possible ones for which there exists an SCN basis     

Image scrambling through a fractional GF(qn) compositedomain

A novel fractional GF(qⁿ) composite domain (FGCD) for image scrambling applications is described. The FGCD scrambled image can be recovered using the appropriate keys without any knowledge of the original image information. Gaussian noise is added to the FGCD transformation results to test the reversing performance of the FGCD     

GF(q)上广义自缩序列的线性复杂度

针对基于GF(q)上m-序列的广义自缩序列,本文利用一种新手段给出线性复杂度上界值.主要讨论素数q大于等于3时,GF(q)上广义自缩序列的线性复杂度.对于GF(3)上广义自缩序列,把以往GF(3)上广义自缩序列的线性复杂度的上界缩小得到一个更精确地上界值.拓展到大于3的素数,给出GF(q)上广义自缩序列的线性复杂度精确...     

Quantum codes from cyclic codes over GF(4m)

We provide a construction for quantum codes (Hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4^m). We also provide examples of these codes some of which meet the known bounds for quantum codes     

Low-density parity check codes over GF(q)

Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10^-5 at E_b/N₀=0.2 dB     

A fast algorithm for determining the linear complexity of asequence with period pn over GF(q)

A fast algorithm is presented for determining the linear complexity of a sequence with period pⁿ over GF (q), where p is an odd prime, and where q is a prime and a primitive root (mod p²)     

Construction of LDPC codes over GF（q） with modified progressive edge growth

A parity check matrix construction method for constructing a low-density parity-check (LDPC) codes over GF(q) (q>2) based on the modified progressive edge growth (PEG) algorithm is introduced. First, the nonzero locations of the parity check matrix are selected using the PEG algorithm. Then the nonzero elements are defined by avoiding the definition of subcode. A proof is given to show the good minimum distance property of constructed GF(q)-LDPC codes. Simulations are also presented to illustrate the good error performance of the designed codes.     

Fast algorithms for determining the linear complexity of sequences over GF(p/sup m/) with period 2/sup t/n

We prove a result which reduces the computation of the linear complexity of a sequence over GF(p^m) (p is an odd prime) with period 2n (n is a positive integer such that there exists an element bisinGF(p^m), bⁿ=-1) to the computation of the linear complexities of two sequences with period n. By combining with some known algorithms such as the Berlekamp-Massey algorithm and the Games-Chan algorithm we can determine the linear complexity of any sequence over GF(p^m) with period 2^tn (such that 2 ^t|p^m-1 and gcd(n,p^m-1)=1) more efficiently     

Relationships between m-sequences over GF(q) andGF(qm)

It is shown that m-sequences over GF(q^m ) of length q^nm-1 corresponding to primitive polynomials in GF[q^m,x] of degree n can be generated from known m-sequences over GF(q) of length q^nm-1 obtained from primitive polynomials in GF[q,x] of degree mn. A procedure for generating the m-sequences over GF(q²) from m-sequences over GF(q) was given which enables the generation of m-sequences over GF( p²ⁿ). In addition it was shown that all of the primitive polynomials in GF[q,^m,x] can be obtained from a complete set of the primitive polynomials in GF[q ,x]     

Fast algorithm for finding primitive polynomials over GF(q)

An efficient algorithm for the construction of primitive polynomials of degree m over GF(q) is proposed. The algorithm runs in time O(km/sup 2/), where k is an integer such that gcd (k,q/sup m-1/)=1.<>     

基于GF(2~n)的ECC协处理器芯片设计

文章讨论了定义在GaloisField(GF)2有限域上椭圆曲线密码体制(ECC)协处理器芯片的设计。首先在详细分析基于GF(2n)ECC算法的基础上提取了最基本和关键的运算,并提出了通过协处理器来完成关键运算步骤,主处理器完成其它运算的ECC加/解密实现方案。其次,进行了加密协处理器体系结构设计,在综合考虑面积、速度、功耗的基础上选择了全串行方案来实现GF(2n)域上的乘和加运算。然后,讨论了加密协处理器芯片的电路设计和仿真、验证问题。最后讨论了芯片的物理设计并给出了样片的测试结果。     

New multilevel codes over GF(q)

Set partitioning is applied to multidimensional signal spaces over GF(q), i.e., GFⁿ¹(q) (n1⩽q ), and it is shown how to construct both multilevel block codes and multilevel trellis codes over GF(q). Multilevel (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum distance d_min⩾d are presented. These codes use Reed-Solomon codes as component codes. Longer multilevel block codes are also constructed using q-ary block codes with block length longer than q+1 as component codes. Some quaternary multilevel block codes are presented with the same length and number of information symbols as, but larger distance than, the best previously known quaternary one-level block codes. It is proved that if all the component block codes are linear. the multilevel block code is also linear. Low-rate q-ary convolutional codes, word-error-correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multilevel trellis codes over GF(q) or binary-to-q-ary trellis codes     

A fast algorithm for determining the minimal polynomial where of a sequence with period 2p/sup n/ over GF (q)

A fast algorithm is presented for determining the linear complexity and the minimal polynomial of a sequence with period 2p/sup n/ over GF (q), where p and q are odd prime, and q is a primitive root (mod p/sup 2/). The algorithm uses the fact that in this case the factorization of x/sup 2p(n)/-1 is especially simple.     

Circulant based extremal additive self-dual codes over GF(4)

It is well known that the problem of finding stabilizer quantum-error-correcting codes (QECC) is transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to classify the extremal additive circulant self-dual codes of lengths up to 15, and construct good codes for lengths 16/spl les/n/spl les/27. We also classify the extremal additive 4-circulant self-dual codes of lengths 4,6,8,12,14, and 16 and most codes of length 10, and construct good codes of even lengths up to 22. Furthermore, we classify the extremal additive bordered 4-circulant self-dual codes of lengths 3,5,7,9,11,13,15, and 17, and construct good codes for lengths 19,21,23, and 25. We give the current status of known extremal (or optimal) additive self-dual codes of lengths 12 to 27.     

大气激光通信系统中GF(17)域RS码研究

低脉冲位置调制是大气激光通信领域广泛采用的调制方式,为了更好地与低脉冲位置调制映射,提高通信系统的检错和纠错能力,在此通信系统中的信道编码采用基于有限域GF(q)上的纠错码,论述了GF(17)域RS码的编译码箅法和研究RS(16,10)的编译码算法及其实现过程.结果表明,采用16脉冲位置调制方式在信道受到干扰后,基于G...     

Generation of m-sequences over GF(q2) from m-sequences over GF(q)

A closed form solution that yields the component shift required for the vector representation of m-sequences over GF(q²) in terms of m-sequences over GF(q²) is presented. Iterative application of this expression then enables the vctor representation of m-sequences over GF(q^2m) in terms of m-sequences over GF(q). These vector m-seqeences can be used directly for the selection of frequencies in frequency hopped spread spectrum communication systems.     

Fast Bit Parallel-Shifted Polynomial Basis Multipliers in GF(2n)

A new nonpipelined bit-parallel-shifted polynomial basis multiplier for GF(2ⁿ) is presented. For some irreducible trinomials, the space complexity of the multiplier matches the best results available in the literature, and its gate delay is equal to T _A+lceillog₂nrceilT_X, where T_A and T_X are the delay of one two-input and and xor gates, respectively. To the best of our knowledge, this is the first time that the gate delay bound T_A+lceillog₂nrceilT_X is reached. For some irreducible pentanomials, its gate delay is equal to T_A +(1+lceillog₂nrceil)T_X. NIST has recommended five binary fields for the elliptic curve digital signature algorithm applications: GF(2¹⁶³), GF(2²³³), GF(2 ²⁸³), GF(2⁴⁰⁹), and GF(2⁵⁷¹), but no irreducible trinomials exist for three degrees, viz., 163, 283 and 571. For the three corresponding binary fields, we show that the gate delay of the proposed multiplier is T_A+(1+lceillog₂nrceil)T_X. This result outperforms the previously known results     

LFSR multipliers over GF(2) defined by all-one polynomial

This paper presents two bit-serial modular multipliers based on the linear feedback shift register using an irreducible all one polynomial (AOP) over GF(2^m). First, a new multiplication algorithm and its architecture are proposed for the modular AB multiplication. Then a new algorithm and architecture for the modular AB² multiplication are derived based on the first multiplier. They have significantly smaller hardware complexity than the previous multipliers because of using the property of AOP. It simplifies the modular reduction compared with the case of using the generalized irreducible polynomial. Since the proposed multipliers have low hardware requirements and regular structures, they are suitable for VLSI implementation. The proposed multipliers can be used as the kernel architecture for the operations of exponentiation, inversion, and division.     

关于q元Alternant码的最小距离

本文给出了ｑ元Ａｌｔｅｒｎａｎｔ码的最小距离一个新的下界，改进了以往有关结果。此外，本文将指出，文献［１］关于Ａｌｔｅｒｎａｎｔ码的最小距离的结果，一般是不成立的。     

New self-dual codes over GF(4) with the highest known minimumweights

The purpose of this correspondence is to construct new Hermitian self-dual codes over GF(4) of lengths 22, 24, 26, 32, and 34 which have the highest known minimum weights. In particular, for length 22, we construct eight new extremal self-dual [22,11,8] codes over GF(4) which do not have a nontrivial automorphism of odd order. The existence of such codes has been left open since 1991 by Huffman