VI-2 class of Hamming weight of -aryq linear codes with dimension 5

The Hamming weight hierarchy of a linear [n,k;q] code c over GF(q)is the sequence(d₁,d₂,…,d_k),where d_r is the smallest support weight of an r-dimensional subcode of c.According to some new necessary conditions,the VI class Hamming weight hierarchies of q -ary linear codes of dimension 5 can be divided into six subclasses. By using the finite projective geometry method, VI-2 subclass and determine were researched almost all weight hierarchies of the VI-2 subclass of weight hierarchies of q -ary linear codes with dimension 5.     

Bounds on the minimum support weights

The minimum support weight, d_r(C), of a linear code C over GF(q) is the minimal size of the support of an r-dimensional subcode of C. A number of bounds on d_r(C) are derived, generalizing the Plotkin bound and the Griesmer bound, as well as giving two new existential bounds. As the main result, it is shown that there exist codes of any given rate R whose ratio d_r/d₁ is lower bounded by a number ranging from (q^r-1)/(q^r -q^r-1) to r, depending on R     

Optimal quaternary linear codes of dimension five

Let d_q(n,k) be the maximum possible minimum Hamming distance of a q-ary [n,k,d]-code for given values of n and k. It is proved that d₄ (33,5)=22, d₄(49,5)=34, d₄ (131,5)=96, d₄(142,5)=104, d₄(147,5)=108, d ₄(152,5)=112, d₄(158,5)=116, d₄(176,5)⩾129, d₄(180,5)⩾132, d₄(190,5)⩾140, d₄(195,5)=144, d₄(200,5)=148, d₄(205,5)=152, d₄(216,5)=160, d₄(227,5)=168, d₄(232,5)=172, d₄(237,5)=176, d₄(240,5)=178, d₄(242,5)=180, and d₄(247,5)=184. A survey of the results of recent work on bounds for quaternary linear codes in dimensions four and five is made and a table with lower and upper bounds for d₄(n,5) is presented     

On the weight hierarchy of Preparata codes over Z4

Hammons et al. (see ibid., vol.40, p.301-19, 1994) showed that, when properly defined, the binary nonlinear Preparata code can be considered as the Gray map of a linear code over Z₄, the so called Preparata code over Z₄. We consider the rth generalized Hamming weight d_r(m) of the Preparata code of length 2^m over Z₄. For any m⩾3, d_r(m) is exactly determined for r=0.5, 1, 1.5, 2, 2.5 and 3.0. For a composite m, we give an upper bound on d_r(m) using the lifting technique. For m=3, 4, 5, 6 and 8, the weight hierarchy is completely determined. In the case of m=7, the weight hierarchy is completely determined except for d₄(7)     

Further results on generalized Hamming weights for Goethals andPreparata codes over Z4

This article contains results on the generalized Hamming weights (GHW) for the Goethals and Preparata codes over Z₄. We give an upper bound on the rth generalized Hamming weights d_r(m,j) for the Goethals code G_m(j) of length 2^m over Z ₄, when m is odd. We also determine d_3.5(m,j) exactly. The upper bound is shown to be tight up to r=3.5. Furthermore, we determine the rth generalized Hamming weight d_r(m) for the Preparata code of length 2^m over Z₄ when r=3.5 and r=4     

Stopping Set Analysis of Iterative Row-Column Decoding of Product Codes

In this paper, we introduce stopping sets for iterative row-column decoding of product codes using optimal constituent decoders. When transmitting over the binary erasure channel (BEC), iterative row-column decoding of product codes using optimal constituent decoders will either be successful, or stop in the unique maximum-size stopping set that is contained in the (initial) set of erased positions. Let C_p denote the product code of two binary linear codes C_c and C_r of minimum distances dc and d_r and second generalized Hamming weights d₂(C_c) and d₂(C_r), respectively. We show that the size s_min of the smallest noncode- word stopping set is at least mm(d_rd₂(C_c),d_cd₂(C_r)) > d_rd_c, where the inequality follows from the Griesmer bound. If there are no codewords in C_p with support set S, where S is a stopping set, then S is said to be a noncodeword stopping set. An immediate consequence is that the erasure probability after iterative row-column decoding using optimal constituent decoders of (finite-length) product codes on the BEC, approaches the erasure probability after maximum-likelihood decoding as the channel erasure probability decreases. We also give an explicit formula for the number of noncodeword stopping sets of size s_min, which depends only on the first nonzero coefficient of the constituent (row and column) first and second support weight enumerators, for the case when d₂(C_r) < 2d_r and d₂(C_c) < 2d_c. Finally, as an example, we apply the derived results to the product of two (extended) Hamming codes and two Golay codes.     

On the second generalized Hamming weight of the dual code of adouble-error-correcting binary BCH code

The generalized Hamming weight of a linear code is a new notion of higher dimensional Hamming weights. Let C be an [n,k] linear code and D be a subcode. The support of D is the cardinality of the set of not-always-zero bit positions of D. The rth generalized Hamming weight of C, denoted by d_r(C), is defined as the minimum support of an r-dimensional subcode of C. It was shown by Wei (1991) that the generalized Hamming weight hierarchy of a linear code completely characterizes the performance of the code on the type II wire-tap channel defined by Ozarow and Wyner (1984). In the present paper the second generalized Hamming weight of the dual code of a double-error-correcting BCH code is derived and the authors prove that except for m=4, the second generalized Hamming weight of [2^m-1, 2m]-dual BCH codes achieves the Griesmer bound     

The weight hierarchies of some product codes

We determine the weight hierarchies of the product of an n-tuple space and an arbitrary code, the product of an m-dimensional even-weight code and the [24,12,8] extended Golay code, and the product of an m-dimensional even-weight code and the [8,4,4] extended Hamming code. The conjecture d_r=d*_r is proven for all three cases     

The q-ary image of a qm-ary cyclic code

For (n, q)=1 V a q^m-ary cyclic code of length n and with generator polynomial g(x), we show that there exists a basis for F(q^m) over F_q with respect to which the q-ary image of V is cyclic, if and only if: (i) g(x) is over F_q; or (ii) g(x)=g₀(x)(x-γ^-q(μ)), g₀(x) is over F_q, F_q≠F(q^k)=F_q(γ)⊂F(q^m ), μ an integer modulo k, and w^m-γ has a divisor over F(q^k) of degree e=m/k; or (iii) g(x)=g₀ (x) Π_μϵs(x-γ(-q^μ)), g ₀(x) is over F_q, F_q≠F(q^k)=F_q(γ)⊂F(q^m ), S a set of integers module k of cardinality k-1 and w^m -μ has a divisor over F(q^k) of degree e=m/k. In all of the above cases, we determine all of the bases with respect to which the q-ary image of V is cyclic     

Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums Sigma_xisinF(q)zeta_p ^Tr(af(x)+bx)(a,bisinF_q,q=p^m,pges3) for all perfect nonlinear functions f which is a Dembowski-Ostrom polynomial or p = 3,f=^x(3(k)+1)/2 where k is odd and (k,m)=1. As applications, we determine (1) the correlation distribution of the m-sequence {a_lambda=Tr(gamma^lambda)}(lambda=0,1,...) and the sequence {b_lambda=Tr(f(gamma^lambda))}(lambda=0,1,...) over F_p where gamma is a primitive element of F_q and (2) the weight distributions of the linear codes over F_p defined by f.     

On MDS extensions of generalized Reed- Solomon codes

An(n, k, d)linear code overF=GF(q)is said to be {em maximum distance separable} (MDS) ifd = n - k + 1. It is shown that an(n, k, n - k + 1)generalized Reed-Solomon code such that2leq k leq n - lfloor (q - 1)/2 rfloor (k neq 3 {rm if} qis even) can be extended by one digit while preserving the MDS property if and only if the resulting extended code is also a generalized Reed-Solomon code. It follows that a generalized Reed-Solomon code withkin the above range can be {em uniquely} extended to a maximal MDS code of lengthq + 1, and that generalized Reed-Solomon codes of lengthq + 1and dimension2leq k leq lfloor q/2 rfloor + 2 (k neq 3 {rm if} qis even) do not have MDS extensions. Hence, in cases where the(q + 1, k)MDS code is essentially unique,(n, k)MDS codes withn > q + 1do not exist.     

Four nonexistence results for ternary linear codes

It is proved that ternary codes with parameters [15,8,6], [15,9,5], [16,6,8], and [16,7,7] do not exist. This result solves the problem of finding optimal ternary linear codes of length at most 21. A table is given, showing the exact value of d₃(n,k) for n⩽21 with the earliest references     

The nonexistence of some five-dimensional quaternary linear codes

Let n₄(k, d) be the smallest integer n, such that a quaternary linear [n, k, d; 4]-code exists. It is proved that n₄ (5, 20)=30, n₄(5, 42)⩾59, n₄(5, 45)⩾63, n₄(5, 64)⩾88, n₄(5, 80)=109, n₄(5, 140)⩾189, n₄(5, 143)⩾193, n₄ (5, 168)⩾226, n₄(5, 180)⩾242, n₄(5, 183)⩾246, n₄(5, 187)=251     

Wide-sense and strict-sense nonblocking operation of multicastmulti-log2 n switching networks

Multicast connections are used in broad-band switching networks as well as in parallel processing. We consider wide-sense and strict-sense nonblocking conditions for multi-log₂ N switching networks with multicast connections. We prove that such networks are wide-sense nonblocking if they are designed by vertically stacking at least t · 2^n-t-1 + 2 ^n-2t-1 planes of a log₂ N networks together, where 1 ⩽ t ⩽ [n/2] and t defines the size of a blocking window K = 2^t. For t = [n/2] and n even, and for [n/2] ⩽ t ⩽ n the number of planes must be at least t · 2^n-t-1 + 1 and 2^t + (n - t - 1) · 2^n-t-1 - 2^2t-n-1 + 1, respectively. In the case of strict-sense nonblocking switching networks, the number of planes is at least N/2. The results obtained in this paper show that in many cases number of planes in wide-sense nonblocking switching networks is less than those for t = [n/2] considered by Tscha and Lee (see ibid., vol.47, p.1425-31, Sept. 1999). The number of planes given in the paper is the minimum number of planes needed for wide-sense nonblocking operation provided that Algorithm 1 is used for setting up connections. The minimum number of planes for such operation in general is still open issue     

New class of nonlinear systematic error detecting codes

A code C detects error e with probability 1-Q(e),ifQ(e) is a fraction of codewords y such that y, y+e/spl isin/C. We present a class of optimal nonlinear q-ary systematic (n, q/sup k/)-codes (robust codes) minimizing over all (n, q/sup k/)-codes the maximum of Q(e) for nonzero e. We also show that any linear (n, q/sup k/)-code V with n /spl les/2k can be modified into a nonlinear (n, q/sup k/)-code C/sub v/ with simple encoding and decoding procedures, such that the set E={e|Q(e)=1} of undetected errors for C/sub v/ is a (k-r)-dimensional subspace of V (|E|=q/sup k-r/ instead of q/sup k/ for V). For the remaining q/sup n/-q/sup k-r/ nonzero errors, Q(e)/spl les/q/sup -r/for q/spl ges/3 and Q(e)/spl les/ 2/sup -r+1/ for q=2.     

The dynamics of group codes: state spaces, trellis diagrams, andcanonical encoders

A group code C over a group G is a set of sequences of group elements that itself forms a group under a component-wise group operation. A group code has a well-defined state space Σ_k at each time k. Each code sequence passes through a well-defined state sequence. The set of all state sequences is also a group code, the state code of C. The state code defines an essentially unique minimal realization of C. The trellis diagram of C is defined by the state code of C and by labels associated with each state transition. The set of all label sequences forms a group code, the label code of C, which is isomorphic to the state code of C. If C is complete and strongly controllable, then a minimal encoder in controller canonical (feedbackfree) form may be constructed from certain sets of shortest possible code sequences, called granules. The size of the state space Σ_k is equal to the size of the state space of this canonical encoder, which is given by a decomposition of the input groups of C at each time k. If C is time-invariant and ν-controllable, then |Σ_k|=Π_1⩽j⩽v|F_j/F _j-1|^j, where F₀ ⊆···⊆ Fν is a normal series, the input chain of C. A group code C has a well-defined trellis section corresponding to any finite interval, regardless of whether it is complete. For a linear time-invariant convolutional code over a field G, these results reduce to known results; however, they depend only on elementary group properties, not on the multiplicative structure of G. Moreover, time-invariance is not required. These results hold for arbitrary groups, and apply to block codes, lattices, time-varying convolutional codes, trellis codes, geometrically uniform codes and discrete-time linear systems     

A New Family of Ternary Almost Perfect Nonlinear Mappings

A mapping f(x) from GF(pⁿ) to GF(pⁿ) is differentially k-uniform if k is the maximum number of solutions x isin GF(pⁿ) of f(x+a) - f(x) = b, where a, b isin GF(pⁿ) and a ne 0. A 2-uniform mapping is called almost perfect nonlinear (APN). This correspondence describes new families of ternary APN mappings over GF(3ⁿ), n>3 odd, of the form f(x) = ux^d + x^{d 2} where d₁ = (3^n-1)/2 - 1 and d₂ = 3ⁿ - 2.     

Some bounds for the minimum length of binary linear codes ofdimension nine

We prove the nonexistence of binary [69,9,32] codes and construct codes with parameters [76,9,34],[297,9,146], and [300,9,148]. These results show that n(9,32)=70, n(9,34)⩽76,n(9,146)=297, and n(9,148)=300, where n(k,d) denotes the smallest value of n for which there exists an [n,k,d] binary code. We also present some codes of minimum distance 32 and some related codes