paper develops the theory and constructions of polyphase zero-ambiguity-zone (ZAZ) sequences for ISAC waveforms, enabling the ZAZ shape to be designed over delay–Doppler regions of interest and supporting flexible (including multi-mode) sensing–communication operation. We first prove that a polyphase sequence with an optimal rectangular auto-ZAZ must be a member of some uncorrelated optimal ZCZ sequence family, and conversely, any member of an uncorrelated optimal ZCZ sequence family has an optimal rectangular auto-ZAZ (Theorems 1, 2 and 3). We generalize the optimality condition on the rectangular ZAZ to that on centrally symmetric convex ZAZs in general (Theorem 4). We propose some constructions of families of polyphase sequences with a strictly or asymptotically optimal rectangular ZAZ (Theorem 1), dual asymptotically optimal rectangular ZAZs (Theorems 5 and 6), and asymptotically optimal rhombic or hexagonal ZAZ (Remarks 5 and 6) from the flexible ZAZ configuration

In this paper, we propose some constructions for the uncorrelated zero-correlation-zone (ZCZ) sequence families over phase shift keying (PSK) / PSK + alphabet as the characteristic sequences of a relative difference set (RDS). The proposed sequence families feature some significant reduction in alphabet size at the price of relaxing the optimality condition, specifically reducing the number of sequences.

In this paper, we propose a simple construction for binary (n,k) linear codes using s-mark Golomb rulers. We prove that these codes are sequential-recovery locally repairable codes (LRCs) with availability 2, which can sequentially recover 5 erased symbols. We prove the necessary and sufficient condition for the proposed codes to be rate-optimal. We also prove the necessary and sufficient condition for the proposed codes to be dimension-optimal. Finally, we propose some variations of this constructions to obtain some 5-sequential recovery LRCs with availability 3. The proposed codes have higher rates and have more flexible choice for the lengths than other previously reported constructions.

This paper proves that Popovi´c's construction describes all the uncorrelated optimal ZCZ sequence families.

In this paper, we define (m,n,r) zero-correlation-zone (ZCZ) sonar sequences and present some of their properties. We prove an upper bound on r for (m,n,r) ZCZ sonar sequences and propose a new and simple construction for (m,n,r) ZCZ sonar sequences with m = r2− 1 and any positive integer n. We also propose two constructions for (m,n,2) ZCZ-DD sonar sequences for some m and n which are some variations of well-known sonar sequence constructions. We report a lot of exhaustive search results and some interesting open problems.

In this paper, we derive a lower bound on the alphabet size of the optimal uncorrelated polyphase ZCZ sequence families, assuming that Mow’s Conjecture is true. We also propose some new optimal uncorrelated polyphase ZCZ sequence families for all the optimal ZCZ parameters. All our proposed families achieve our proposed alphabet size bound so that this bound is the minimum alphabet size (assuming Mow’s conjecture is true). We also derive some interesting facts regardless of the truth of Mow’s Conjecture: an optimal ZCZ family of normalized sequence (not necessarily uncorrelated) is always periodic complementary; therefore, a sequence generated by interleaving all the sequences in an optimal uncorrelated ZCZ family of normalized sequences is always a perfect sequence.

We define run sequences of period$2^{n}-1$as the binary sequences where the distribution of runs of 0’s and runs of 1’s is exactly same as that for the maximal length linear shift resister sequences of period$2^{n}-1$. We first count the number of all the cyclically distinct run sequences of period$2^{n}-1$. For each$n$-tuple, we consider the average number of occurrences over all the run sequences of period$2^{n}-1$. We identify the$n$-tuples with average number 1 and, in particular, those that occur exactly once in every run sequence of period$2^{n}-1$. We finally prove that, as$n$increases, the average number of every non-zero$n$-tuple approaches to 1.

This paper briefly introduces the transmitter of the GNSS signal to which CSK modulation is applied, and summarizes the CSK hard/soft decision demodulation method in the corresponding receiver. Based on the summarized method, we show the hard/soft decision demodulation performance and various analyzes in an environment similar to the existing GNSS signal.

In this paper, we analyzes performance of girth-8 regular QC-LDPC codes constructed using Golomb ruler. We conducted simulations to measure FER performance of QC-LDPC codes constructed by changing the last mark of some optimal Golomb ruler and found the value of the mark that shows the best performance.

Sidelnikov sequences, a kind of cyclotomic sequences with many desired properties such as low correlation and variable alphabet sizes, can be employed to construct a polyphase sequence family that has many applications in high-speed data communications. Recently, cyclotomic numbers have been used to investigate the linear complexity of Sidelnikov sequences, mainly about binary ones, although the limitation on the orders of the available cyclotomic numbers makes it difficult. This paper continues to study the linear complexity over$\mathbb {F}_{M}$of$M$-ary Sidelnikov sequence of period$q-1$using Hasse derivative, which implies$q=p^{m}$,$m\geq 1$and$M|(q-1)$. The$t$th Hasse derivative formulas are presented in terms of cyclotomic numbers, and some upper bounds on the linear complexity for$M=2$and 3 are obtained only with some additional restrictions on$q$. Furthermore, concrete illustrations for several families of these sequences, such as$q\equiv 1\pmod {2}$and$q\equiv 1\pmod {3}$, show these upper bounds are tight and reachable; especially for$q=2\times 3^{\lambda }+1 (1\leq \lambda \leq 20)$, the exact linear complexities over$\mathbb {F}_{3}$of the ternary Sidelnikov sequences are determined; and it turns out that all the linear complexities of the sequences considered are very close to their periods.

In this paper, we give a new framework for constructing perfect sequences, called generalized Milewski sequences, over various alphabets including Polyphase (PSK) as well as Amplitude-and-Polyphase (APSK) in general, and for constructing optimal sets of such perfect sequences by using combinatorial designs, called circular Florentine arrays. Specifically, we prove that, given any positive integer$m\geq 1$, (i) there exists a perfect sequence of period$mN^{2}$for any positive integer$N$if there exists a perfect sequence (polyphase or not) of length$m$; (ii) an optimal$k$-set of perfect sequences of length$mN^{2}$can be constructed if there exist both a$k \times N$circular Florentine array and an optimal$k$-set of perfect sequences all of length$m$. This enables us to find some optimal$k$-set of perfect sequences where$k > p_{\text {min}}-1$, where$p_{\text {min}}$is the smallest prime factor of$mN^{2}$.

In this paper, we calculate the number of sequences with run property and propose two methods for generating sequences with run property. One of them, generating sequences in order, is useful for exhaustive search, and another, ranking and unranking method, is useful for scatter search.

The linear complexity is a measure for the unpredictability of a sequence over a finite field. Sequences with good pseudo-random properties and large linear complexity are widely used in the CDMA spread spectrum communication and cryptography. In recent years, many researchers have focused on the linear complexity of cyclotomic sequences such as Sidel'nikov sequence. This paper studies the FM-linear complexity of M-ary Sidel'nikov sequence of period p-1 using the Hasse derivative of its generating function, where M|(p-1). The tth Hasse derivative formulas are generalized in terms of cyclotomic numbers, and then the exact F3-linear complexities of the ternary Sidel'nikov sequences are determined for p = 2·3λ+1(1 ≤ λ ≤ 20). It turns out that all of the linear complexities of the considered sequences are very close to their periods.

In this paper, we propose construction of Reed-Solomon(RS) based Quasi-Cyclic Low-Density Parity-Check(QC-LDPC) codes using protograph combining two existing QC-LDPC codes construction. One is the construction RS based QC-LDPC codes whose girth is at least 8 and another is protograph based QC-LDPC codes to increase the upper bounds of minimum Hamming distance. We construct the protographs to increase the upper bound of minimum Hamming distance for Proto-RS-QC-LDPC codes and simulate some experiment to show coding gain in sense of BER compare to existing QC-LDPC codes.

We analyze the symbol measures for iterative erasure insertion and decoding of a Reed-Solomon coded SFH/MFSK system over jamming channels. In contrast to conventional erasure insertion schemes, iterative schemes do not require any preoptimized threshold or channel state information at the receiver. We confirm the performance improvement using a generalized minimum distance (GMD) decoding method with three different symbol measures. To analyze performance, we propose a new analysis framework considering the “trapped-error” probability. From analysis and the simulation results, we show that ratio-based GMD decoding has the best performance among the one-dimensional iterative erasure insertion and decoding schemes.

In this paper, we give a construction of optimal families of N-ary perfect sequences of period N2, where N is a positive odd integer. For this, we re-define perfect generators and optimal generators of any length N which were originally defined only for odd prime lengths by Park, Song, Kim, and Golomb in 2016, but investigate the necessary and sufficient condition for these generators for arbitrary length N. Based on this, we propose a construction of odd length optimal generators by using odd prime length optimal generators. For a fixed odd integer N and its odd prime factor p, the proposed construction guarantees at least (N/p)p-1φ(N/p)φ(p)φ(p-1)/φ(N)2inequivalent optimal generators of length N in the sense of constant multiples, cyclic shifts, and/or decimations. Here, φ(·) is Euler's totient function. From an optimal generator one can construct lots of different N-ary optimal families of period N2, all of which contain pmin-1 perfect sequences, where pminis the least positive prime factor of N.

In this paper, we propose a new construction of perfect pk-ary sequences of period pk, where p is an odd prime and k ≥ 2 is a positive integer, based on cubic polynomials over the integers modulo pk. We show that, for some appropriate parameters, it generates perfect polyphase sequences which are not the generalized chirp-like sequences constructed by Popovic in 1992.

In this paper, we show that the cross-correlation of two properly chosen column sequences from the array structure of two different Sidelnikov sequences of periods qe-1 and qf- 1, where e ≠ f, is bounded by (e+f-1)√q+1. From this result, we construct new sequence families by combining sequence families from the array structure of Sidelnikov sequences of period q2- 1, q3- 1,..., qd- 1 for some d with 2 ≤ d ≤ 1/2(√q - 2/√q + 1). The maximum non-trivial complex correlation of any two pair of sequences in the constructed sequence family is upper-bounded by (2d - 1)√q + 1: thus, the combining process does not affect the maximum non-trivial complex correlation.

We show that a p-ary polyphase sequence of period p2from the Fermat quotients is perfect. That is, its periodic autocorrelation is zero for all non-trivial phase shifts. We call this Fermat-quotient sequence. We propose a collection of optimal families of perfect polyphase sequences using the Fermatquotient sequences in the sense of the Sarwate bound. That is, the cross correlation of two members in a family is upper bounded by p. To investigate some relation between Fermat-quotient sequences and Frank-Zadoff sequences and to construct optimal families including these sequences, we introduce generators of p-ary polyphase sequences of period p2using their p × p array structures. We call an optimal generator to be the generator of some p-ary polyphase sequences which are perfect and which gives an optimal family by the proposed construction. Finally, we propose an algebraic construction for optimal generators as another main result. A lot of optimal families of size p - 1 can be constructed from these optimal generators, some of which are known to be from the Fermat-quotient sequences or from the Frank-Zadoff sequences, but some families are new for p ≥ 11. The relation between the Fermat-quotient sequences and the Frank-Zadoff sequences is determined as a by-product.

We show that a p-ary polyphase sequence of period p2from the Fermat quotients is `perfect.' That is, its periodic autocorrelation is zero for all non-trivial shifts. We call this Fermat-Quotient sequences. Using this and the fact that the Frank-Zadoff sequences (which is known to be also perfect), we propose a collection of `optimum' families of perfect polyphase sequences in the sense of Sarwate Bound. That is, the cross-correlation of two members in a family is upper bounded by p. We may say these families are `completely optimum' since the cross-correlation of any two members in a family is exactly p for all phase-shifts.

In this paper, we extend the construction by Vu and Gong for families of M-ary sequences of period q - 1 from the array structure of an M-ary Sidelnikov sequence of period q2- 1, where q is a prime power and M|q - 1. The construction now applies to the cases of using any period qd-1 for 3 ≤ d27. The proposed construction results in a family of M-ary sequences of period q-1 with: 1) the correlation magnitudes, which are upper bounded by (2d -1)√q +1 and 2) the asymptotic size of (M -1)qd-1/d as q increases. We also characterize some subsets of the above of size ~(r - 1)qd-1/d but with a tighter upper bound (2d - 2)√q + 2 on its correlation magnitude. We discuss reducing both time and memory complexities for the practical implementation of such constructions in some special cases. We further give some approximate size of the newly constructed families in general and an exact count when d is a prime power or a product of two distinct primes. The main results of this paper now give more freedom of tradeoff in the design of M-ary sequence family between the family size and the correlation magnitude of the family.

In this paper, we propose an optimization scheme for degree distributions of LDPC codes that combines differential evolution and iterative simplex algorithm. In the proposed scheme, we find good check and variable nodes degree distribution and threshold by using differential evolution, and then, we further optimize check and variable nodes degree distribution that has enhanced code rate by using an iterative simplex algorithm. An iterative simplex algorithm consists of two simplex algorithms, optimizing ρ(x) and λ(x), respectively and iteratively. Some simulation results show that the proposed scheme finds degree distributions, better than those obtained by the differential evolution only, in terms of threshold and/or code rate. As an application of the proposed scheme, we report some new degree distributions with some new maximum degrees.

In this paper, we present an analysis on the performance of Plotkin-type UEP codes by using threshold and equivalent channel model under the Gaussian assumption. Specifically, we give a framework on how to assign the rate of each component code of this Plotkin-type UEP code so that the goal of UEP is successfully achieved with far better overall BER performance. The result is surprising in that which component code should be assigned for MSB can be changed according to the assigned rate of each component code and overall rate.

This paper deals with the global optimality of the channel capacity of the multiple-input multiple-output (MIMO) cooperative system which is equipped with precoders at source and relay, and exploits the direct channel between source and destination. Each precoder of the system is individually designed by the Lagrangian method and the final precoders are decided by an iterative structure. To prove the global optimality for the channel capacity of the system with these joint precoders, we show that the channel capacity function of the system is a concave function and the constraints of the system are convex sets.

Let$p=ef+1$be an odd prime for some$e$and$f$, and let$F_{p}$be the finite field with$p$elements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period$p$) of all the cyclic difference sets$D$which are some union of cosets of$e$th powers$H_{e}$in$F_{p}^{\ast }(\triangleq F_{p}\backslash \{0\})$for$e\leq 12$. For this, we define$e$th power residue sequences of period$p$, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of$H_{e}$in$F_{p}^{\ast }$at some primitive$p$th root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known$e$th residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant$e$th power residue sequence for any$e$to be either$p-1$or$p$whenever$(e,(p-1)/n)=1$, where$n$is the order of 2 mod$p$.

Let$p$be a prime and$n$a positive integer. Let$e\vert p^{n}-1$and$N={{p^{n}-1}\over{e}}$. In this paper, we construct a family$S$of$e^{2}N~p$-ary sequences, each member of$S$has period$N$and the magnitudes of correlations of members of$S$are upper bounded by$2\sqrt{p^{n}}=2\sqrt{eN+1}$.

We apply the Hadamard equivalence to all the binary matrices of size m × n and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class and count the number of HR-minimals of size m × n for m ≤ 3. Some properties and constructions of HR-minimals are investigated. HR-minimals with the largest weight on its second row are defined as Quasi-Hadamard matrices, which are very similar to Hadamard matrices in terms of the absolute correlations of pairs of rows, in the sense that they give a set of row vectors with “best possible orthogonality.” We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.

We investigate properties of a pair of binary sequences having ideal two-level crosscorrelation function. The concept of the associated cyclic difference pair is given. Equivalent descriptions via polynomial representation and circulant matrix are given. For a period of every multiple of 4, there exists a binary sequence pair whose in-phase correlation is 4 and all out-of-phase correlations are zero. Based on the result of exhaustive computer search for short lengths we conjecture that the construction given exhausts all possible classes and is unique up to correlation preserving transformations.

We review prime n-square sequences of length pnwhich is originally defined by Ding and Helleseth in 1998, where p is an odd prime and n is a positive integer. In this paper, we determine the linear complexity and the minimal polynomial of these sequences for any n. It turned out that these sequences have linear complexity that is of the order of the period pn.

We propose a modification on the improved progressive-edge-growth(IPEG) algorithm. Proposed modification increases the connectivity of variable nodes using extrinsic message degree of variable nodes, which results in reducing the small stopping sets. Through computer simulation, we confirm that the codes constructed by the proposed algorithm have lower error floor than those constructed by the original IPEG algorithm.

In the parallel decoding of turbo codes, the constituent interleaver must avoid the memory collision. This paper proposes a collision-free interleaver structure which can be optimized easily over various information blocksizes. The performance of the proposed interleaver is almost the same as or 0.1dB loss against almost regular permutation (ARP) at FER 10-5region with information block sizes of 320 and 640 when the simulation environment is given by the 3GPP standard turbo codes with 4 parallelism in AWGN channel.

In this correspondence, we present a connection between designing low-correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors. This results in LCZ signal sets with huge sizes over three different alphabetic sets: finite field of size$q$, integer residue ring modulo$q$, and the subset in the complex field which consists of powers of a primitive$q$th root of unity. We show a connection between these sequence designs and “completely noncyclic” Hadamard matrices and a construction for those sequences. We also provide some open problems along this direction.

In this correspondence, we prove that the complex cross-correlation of a k-ary Sidel'nikov sequence of period q-1 and its constant multiple sequence is upper bounded by radicq+3, where q=pmand here p is an odd prime and m is a positive integer

Let p be an odd prime, q be a divisor of p - 1 and mu be a primitive root mod p. A g-ary PRS (power residue sequence) of period p is defined as s(n) = k if n isin Ckwhere Ck= {muqt+k|t = 0,1,2,...,T - 1} where T = (p - 1)/q. In this paper, we prove that the maximum absolute value of the periodic crosscorrelation of two distinct q-ary PRS's of period p is upper bounded by √ p + 2

We propose a design criterion for serially concatenated LDGM codes which require single decoder. The inner LDGM code can be obtained by expanding the rows of the parity check matrix of the outer LDGM code. The resulting codes can be decoded using only the inner LDGM decoder with slight modification. Simulation results show that the performance of the proposed codes is almost the same as that of serially concatenated LDGM codes's using both the inner and the outer decoder.

We derive the general power spectral density of impulse radio (IR) ultra-wide bandwidth (UWB) systems using a pulse position modulation. We propose a new IR-UWB system with a preferable line spectrum property and a bit error rate performance.

We characterize pk-ary (p prime, k integer) generalized m-sequences and generalized GMW sequences of period p2k-1 over a residue class ring R=GF(p)[xi]/(xik) with the "strictly" optimal partial Hamming autocorrelation function (HAF)

We consider a design of quasicyclic LDPC codes which have irregular column weights. We derive bit-oriented bound and parity-oriented bound on the minimum distances of irregular quasicyclic codes using a similar technique to Tanner's simple lower bound

We classify some p/sup k/-ary (p prime, k integer) generalized m-sequences and generalized Gordon-Mills-Welch (GMW) sequences of period p/sup 2k/-1 over a residue class ring R=GF(p)[/spl xi/]/(/spl xi//sup k/) having optimal partial Hamming autocorrelation properties. In frequency hopping (FH) spread-spectrum systems, these sequences are useful for synchronizing process. Suppose, for example, that a transmitting p/sup k/-ary FH patterns of period p/sup 2k/-1 are correlated at a receiver. Usually, the length of a correlation window, denoted by L, is shorter than the pattern's overall period. In that case, the maximum value of the out-of-phase Hamming autocorrelation is lower-bounded by /spl lceil/L/p/sup k/+1/spl rceil/ but the classified sequences achieve this bound with equality for any positive integer L.

In this correspondence, the characteristic polynomial and hence the linear complexity of Hall's sextic residue sequences are determined.

We present a construction for binary sequences {s(t)} of period N=p/sup m/-1 for an odd prime p based on the polynomial (z+1)/sup d/+az/sup d/+b, and discuss them in some cases of parameters p, m, d, a, and b. We show that new sequences from our construction are balanced or almost balanced and have optimal three-level autocorrelation for the case when the polynomial (z+1)/sup d/+z/sup d/+a can be transformed into the form z/sup 2/-c. We also derive the distribution of autocorrelation values they take on. The sequences satisfy constant-on-the-coset property, and we show that there are more than one characteristic phases with constant-on-the-coset property. Some other interesting properties of those sequences are presented. For the cases when the polynomial (z+1)/sup d/+z/sup d/+a cannot be transformed into the form z/sup 2/-c, we performed extensive computer search, and results are summarized. Based on these results, some open problems are formulated.

It is important to use a better criterion in selection and discretization of attributes for the generation of decision trees to construct a better classifier in the area of pattern recognition in order to intelligently access huge amount of data efficiently. Two well-known criteria are gain and gain ratio, both based on the entropy of partitions. We propose in this paper a new criterion based also on entropy, and use both theoretical analysis and computer simulation to demonstrate that it works better than gain or gain ratio in a wide variety of situations. We use the usual entropy calculation where the base of the logarithm is not two but the number of successors to the node. Our theoretical analysis leads some specific situations in which the new criterion works always better than gain or gain ratio, and the simulation result may implicitly cover all the other situations not covered by the analysis.

We present a construction, in a closed form, for an optimal family of 2/sup m/ binary sequences of period 2/sup 2m/-1 with respect to Welch's (1974) bound, whenever there exists a balanced binary sequence of period 2/sup m/-1 with ideal autocorrelation property using the trace function. This construction enables us to reinterpret a small set of Kasami and No (1988) sequences as a family constructed from m-sequences. New optimal families of binary sequences are constructed from the Legendre sequences of Mersenne prime period, Hall's sextic residue sequences, and miscellaneous sequences of unknown type. In addition, we enumerate the number of distinct families of binary sequences, which are constructed from a given binary sequence by this method.

A hierarchical wavelet transform coding structure which satisfies the requirements of a high definition video cassette recorder (HD-VCR) for studio application in wireless environments is proposed. All of the coefficients in the transformed image are adaptively quantized and hierarchically converted into a single stream or binary symbols using the monotone decreasing property. The resulting bit stream goes through an adaptive arithmetic coder, and two-level control of the bit rate is performed so as to maintain a constant target bit rate. The proposed algorithm can be efficiently implemented with much reduced computing time for the real world application. It is verified by simulation that for a test sequence of images at least 4 dB of PSNR improvement can be achieved compared with other DCT-based schemes at a constant target bit rate, avoiding the blocking effect which would be impossible otherwise.

In this correspondence, it is shown that Legendre sequences of period p can be explicitly represented using the trace function defined on the finite field with 2/sup n/ elements, whenever p=2/sup n/-1 is prime for some n/spl ges/3.

Three constructions for n-dimensional regular simplex codes /spl alpha//sub i/, 0/spl les/i/spl les/n, are proposed, two of which have the property that /spl alpha//sub i/ for 1/spl les/i/spl les/n is a cyclic shift of /spl alpha//sub 1/. The first method is shown to work for all the positive integers n=1,2,... using only three real values. It turns out that these values are rational whenever n+1 is a square of some integer. Whenever a (v,k,/spl lambda/) cyclic (or Abelian) difference set exists, this method is generalized so that a similar method is shown to work with /spl nu/=n (the number of dimensions).>

The main conjecture of this article is the following: if a cyclic (v=4n-1, k=2n-1, /spl lambda/=n-1) Hadamard difference set exists, the the value of v must be either a prime, or a product of "twin primes," or one less than a power of 2. Six cases, v=399, 495, 627, 651, 783, and 975, which were once listed as the possible exceptions for v>

Picking up exactly one member from each of the nonperiodic cyclic equivalence classes of an (n, k+1) Reed-Solomon code E over GF(q) gives a code, E", which has bounded Hamming correlation values and the self-synchronizing property. The exact size of E" is shown to be (1/n) Sigma /sub d mod n/ mu (d)q/sup 1+k/d/, where mu (d) is the Mobius function, (x) is the integer part of x, and the summation is over all the divisors d of n=q-1. A construction for a subset V of E is given to prove that mod E" mod >or= mod V mod =(q/sup k+1/-q/sup k+1-N/)/(q-1) where N is the number of integers from 1 to k which are relatively prime to q-1. A necessary and sufficient condition for mod E" mod = mod V mod is proved and some special cases are presented with examples. For all possible values of q>2, a number B(q) is determined such that mod E" mod = mod V mod for 1mod V mod for k>B(q).>