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     

On (k, t)-subnormal covering codes

The concept of a (k, t)-subnormal covering code is defined. It is discussed how an amalgamated-direct-sumlike construction can be used to combine such codes. The existence of optimal (q, n, M) 1 codes C is discussed such that by puncturing the first coordinate of C one obtains a code with (q, 1)-subnorm 2     

A coding scheme for m-out-of-n codes

A scheme for the construction of m-out-of-n codes based on the arithmetic coding technique is described. For appropriate values of n, k, and m, the scheme can be used to construct an (n,k) block code in which all the codewords are of weight m. Such codes are useful, for example, in providing perfect error detection capability in asymmetric channels such as optical communication links and laser disks. The encoding and decoding algorithms of the scheme perform simple arithmetic operations recursively, thereby facilitating the construction of codes with relatively long block sizes. The scheme also allows the construction of optimal or nearly optimal m-out-of-n codes for a wide range of block sizes limited only by the arithmetic precision used     

Identities and approximations for the weight distribution of q-ary codes

An explicit formula is derived that enumerates the complete weight distribution of an (n, k, d) linear code using a partially known weight distribution. An approximation formula for the weight distribution of q-ary linear (n, k , d) codes is also derived. It is shown that, for a given q-ary linear (n, k, d) code, the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q=q ^-(n-k) as u becomes large. The error term is a decreasing function of the minimum weight of the dual. The results are also valid for nonlinear (n, M, d) codes with the minimum weight of the dual replaced by the dual distance     

On the cutoff rate of a discrete memoryless channel with(d, k)-constrained input sequences

The cutoff rate of a discrete memoryless channel whose output sequences are from a (d, k) encoder is investigated. A rational rate (d, k) encoder is considered as a finite state machine and maximum-likelihood decoding is used to compute the cutoff rate. Some commonly used (d, k) codes, such as the rate 1/2 (1, 3) code with a two-state encoder, the IBM rate 2/3 (1, 7) code having a five-state encoder, and the IBM rate 1/2 (2, 7) code with a seven-state encoder, are used to illustrate the cutoff rate computation. Results are presented for both the binary symmetric channel (BSC) and the Gaussian noise channel. The performance of a decoder designed for noiseless transmission of (1, 3) code is compared to that of a maximum-likelihood decoder for the (1, 3) code. It is also shown that for the case of the Gaussian noise channel, a gain of about 1.7 dB in signal-to-noise ratio is possible by using 3-bit soft decisions over hard decisions     

Binary linear quasi-perfect codes are normal

Whether quasi-perfect codes are normal is addressed. Let C be a code of length n, dimension k, covering radius R, and minimal distance d. It is proved that C is normal if d⩾2R-1. Hence all quasi-perfect codes are normal. Consequently, any [n,k ]R binary linear code with minimal distance d⩾2R-1 is normal     

Generalized Hamming weights of linear codes

The generalized Hamming weight, d_r(C), of a binary linear code C is the size of the smallest support of any r-dimensional subcode of C. The parameter d_r(C) determines the code's performance on the wire-tap channel of Type II. Bounds on d_r(C), and in some cases exact expressions, are derived. In particular, a generalized Griesmer bound for d_r(C) is presented and examples are given of codes meeting this bound with equality     

Binary [18,11]2 codes do not exist-Nor do [64,53]2 codes

A binary, linear block code C with block length n and dimension n is commonly denoted by [n, k] or, if its minimum distance is d, by [n, k,d]. The code's covering radius r(C) can be defined as the smallest number r such that any binary column vector of length (n-k) can be written as a sum of r or fewer columns of a parity-check matrix of C. An [n,k] code with covering radius r is denoted by [n,k]r. R.A. Brualdi et al., (1989) showed that l(m,r) is defined to be the smallest n such that an [n,n-m]r code exists. l(m,2) is known for m⩽6, while it is shown by Brualdi et al. that 17⩽l(7,2)⩽19. This lower bound is improved by A.R. Calderbank et al. (1988), where it is shown that [17,10]2 codes do not exist. The nonexistence of [18,11]2 codes is proved, so that l(7,2)=19. l[7.2)=19 is established by showing that [18,11]2 codes do not exist. It is also shown that [64,53]2 codes do not exist, implying that l(11,2)⩾65     

On linear structure and phase rotation invariant properties ofblock M-PSK modulation codes

Two important structural properties of block M(=2^' )-ary PSK modulation codes, linear structure and phase symmetry, are investigated. An M-ary modulation code is first represented as a code with symbols from the integer group S_M-PSK=(0,1,2,---,M-1) under modulo-M addition. Then the linear structure of block M-PSK modulation codes over S_M-PSK with respect to modulo- M vector addition is defined, and conditions are derived under which a block M-PSK modulation code is linear. Once the linear structure is developed, the phase symmetry of block M-PSK modulation codes is studied. In particular, a necessary and sufficient condition for a block M-PSK modulation code that is linear as a binary code to be invariant under 2^h/180°M phase rotation, for 1⩽h⩽l is derived. Finally, a list of short 8-PSK and 16-PSK modulation codes is given, together with their linear structure and the smallest phase rotation for which a code is invariant     

On the weight distribution of linear codes having dual distanced'⩾k

Using only the principle of inclusion and exclusion, the author derives a formula for the weight distribution of an [n,k ] code whose dual code has a minimum distance d'⩾k . The result yields a new condition on the weight distributions of a linear code and its dual which is necessary and sufficient for the code to be a maximum distance separable (MDS) code. Moreover, it shows how the weight distribution for linear MDS codes is obtained in an elementary manner     

Constructions of error-correcting DC-free block codes

Two DC-free codes are presented with distance 2d, b ⩾1 length 2n+2r(d-1) for d⩽3 and length 2n+2r(d-1)(2d -1) for d>3, where r is the least integer ⩾log₂ (2n+1). For the first code l=4, c=2, and the asymptotic rate of this code is 0.7925. For the second code l=6, c=3, and the asymptotic rate of this code is 0.8858. Asymptotically, these rates achieve the channel capacity. For small values of n these codes do not achieve the best rate. As an example of codes of short length with good rate, the author presents a (30, 10, 6, 4) DC-free block code with 2²¹ codewords. A construction is presented for which from a given code C ₁ of length n, even weight, and distance 4, the author obtains a (4n, l, c, 4) DC-free block code C₂, where l is 4, 5 or 6, and c is not greater than n+1 (but usually significantly smaller). The codes obtained by this method have good rates for small lengths. The encoding and decoding procedures for all the codes are discussed     

Runlength-limited codes for mixed-error channels

The mixed-error channel (MC) combines the binary symmetric channel and the peak shift channel. The construction of (d, k) constrained t-MC-error-correcting block codes is described. It is demonstrated that these codes can achieve a code rate close to the ( d, k) capacity. The encoding and decoding procedures are described. The performance of the construction depends on a particular partitioning of (d, k) constrained block codes. This partitioning is discussed and various tables of codes are included. Examples on encoding/decoding and on code performance are given     

Gain measurements on the KrF(B→X), XeF(B→X), and XeF(C→A) lasertransitions in an XeF(C→A) laser gas mixture

Optimum energy extraction from an electron-beam-pumped XeF(C →A) laser is achieved with a five-component rare gas halide mixture. The characterization and modeling of laser action in such a gas mixture requires a knowledge of small-signal gain and absorption coefficients not only on the blue-green XeF(C→ A) transition, but also in the ultraviolet (UV) region for the competing XeF(B→X) and KrF(B→X ) transitions. The authors report gain measurements on the XeF(C→A) transition and small-signal gain and absorption coefficients at or near both the XeF(B→X ) (351 and 353 nm) and KrF(B→X) (248 nm) transitions. A study of the gain for the UV and visible transitions as a function of Kr and Xe partial pressure is reported, and its impact on the XeF(C→A) kinetics is discussed     

Optimum codes of dimension 3 and 4 over GF(4)

n_q(k,d), the length of a q-ary optimum code for given k and d, for q=4 and k=3, 4 is discussed. The problem is completely solved for k=3, and the exact value of n₄(4,d) is determined for all but 52 values of d     

Coset correlation of m-sequences

It is well-known that there exists a unique shift l of the m-sequence S(k) such that the value of S₀(k)=S(k+l) is only determined by the cyclotomic coset to which k belongs. A measure called the `coset correlation' is introduced. It is proven that the shift l can be determined by the coset correlation     

New minimum distance bounds for certain binary linear codes

Let an [n, k, d]-code denote a binary linear code of length n, dimension k, and minimum distance at least d. Define d(n, k) as the maximum value of d for which there exists a binary linear [n, k, d]-code. T. Verhoeff (1989) has provided an updated table of bounds on d(n, k) for 1⩽k⩽n⩽127. The authors improve on some of the upper bounds given in that table by proving the nonexistence of codes with certain parameters     

Scaling characteristics of the XeF (C→A)excimer laser

The scaling characteristics and medium properties of an injection-controlled XeF(C→A) laser pumped by a 10-ns-high current density electron beam have been investigated. A five-component laser gas mixture, consisting of F₂, NF₃ , Xe, Kr, and Ar was optimized for the scaled laser conditions, resulting in 0.8-J output pulses at 486.8 nm, corresponding to an energy density of small-signal-gain measurements combined with kinetic modeling permitted the characteristics of the dependence of net gain on the electron-beam energy deposition and gas mixture composition, resulting in an improved understanding of XeF(C→A) laser operation     

Closest coset decoding of |u|u+v| codes

The general concept of closest coset decoding (CCD) is presented, and a soft-decoding technique for block codes that is based on partitioning a code into a subcode and its cosets is described. The computational complexity of the CCD algorithm is significantly less than that required if a maximum-likelihood detector (MLD) is used. A set-partitioning procedure and details of the CCD algorithm for soft decoding of |u|u+v| codes are presented. Upper bounds on the bit-error-rate (BER) performance of the proposed algorithm are combined, and numerical results and computer simulation tests for the BER performance of second-order Reed-Muller codes of length 16 and 32 are presented. The algorithm is a suboptimum decoding scheme and, in the range of signal-to-noise-power-density ratios of interest, its BER performance is only a few tenths of a dB inferior to the performance of the MLD for the codes examined     

On the influence of coding on the mean time to failure fordegrading memories with defects

The application of a combined test-error-correcting procedure is studied to improve the mean time to failure (MTTF) for degrading memory systems with defects. The degradation is characterized by the probability p that within a unit of time a memory cell changes from the operational state to the permanent defect state. Bounds are given on the MTTF and it is shown that, for memories with N words of k information bits, coding gives an improvement in MTTF proportional to (k/n) N(d_min^-2)/(d_min ^-1), where d_min and (k/n) are the minimum distance and the efficiency of the code used, respectively. Thus the time gain for a simple minimum-distance-3 is proportional to N^-1. A memory word test is combined with a simple defect-matching code. This yields reliable operation with one defect in a word of length k+2 at a code efficiency k/(k+2)     

On runlength codes

Several results on binary (d, k) codes are given. First, a novel derivation for the capacity of these codes based on information-theoretic principles is given. Based on this result the spectrum of a (d, k) code is computed. Finally, the problem of computing the capacity of the binary symmetric channel under the condition that the input sequences satisfy the (d, k ) constraint is considered. Lower bounds on the capacity of such a channel are derived