Let Z4denote the ring of integers modulo 4. The Galois ring GR(4,m), which consists of 4melements, represents the Galois extension of degreemover Z4. The constructions of codes over Z4have garnered significant interest in recent years. In this paper, building upon previous research, we utilize the defining-set approach to construct several classes of linear codes over Z4by effectively using the properties of the trace function from GR(4,m) to Z4. As a result, we have been able to obtain new infinite families of linear codes over Z4and completely determine their Lee weight distributions.

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over Fqare also obtained, whereqis a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case ofq= 2; 3; 4. It is noteworthy that most of the derived codes over Fqsatisfy the Ashikhmin-Barg’s condition for minimality. Besides, we obtain two infinite families of distanceoptimal codes over Fqwith respect to the Griesmer bound. By puncturing the Gray image codes and subfield-like codes, several classes of projective codes are presented.

Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the minimal length of feedback shift registers that can generate a given sequence. This paper establishes the structure ofn-periodic binary sequences with nonlinear complexity larger than or equal to b [3n/4] Based on their structure, an exact enumeration formula for the number of such periodic sequences is determined.

Constant amplitude zero auto-correlation (CAZAC) sequences are widely applied in waveforms for radar and communication systems. Motivated by a recent work [Berggren and Popovi´c, IEEE Trans. Inf. Theory 70(8), 6068-6075 (2024)], this paper advances the approach to generating CAZAC sequences by interleaving Zadoff-Chu (ZC) sequences with permutation polynomials (PPs). We propose one class of high-degree PPs over the integer ring ZN, and utilize them and their inverses to interleave ZC sequences for constructing CAZAC sequences. It is known that a CAZAC sequence can be extended to an equivalence class by five basic operations. We further show that the obtained CAZAC sequences are not covered by the equivalence classes of ZC sequences and interleaved ZC sequences by quadratic PPs and their inverses, and prove the sufficiency of the conjecture by Berggren and Popovi´c in the aforementioned work. In addition, we also evaluate the aperiodic auto-correlation of certain ZC sequences interleaved by quadratic PPs.

Abstract Differential-linear (DL) cryptanalysis divides the target cipher $E$ into three part, i.e. $E = E_{2} \circ E_{m} \circ E_{1}$. Existing DL distinguishers search frameworks typically begin by estimating the theoretical correlation of $E_{m}$, followed by an experimental evaluation to determine its precise value. However, the deviation between the actual correlation and the theoretical correlation often renders the distinguishers identified by the models invalid. In this paper, we propose a pre-pruning technique to reduce the frequency of invalid distinguishers and improve the existing Mixed-Integer Linear Programming (MILP)-based DL distinguishers search frameworks. Specifically, we first filter the output differences of $E_{d}$ according to the probability of one-round differential characteristics. Subsequently, we identify the high-correlation bits of the output mask of the middle part and designate the low-correlation bits as inactive mask bits in our MILP models for each selected difference. Our pre-pruning technique significantly reduces the number of low-correlation distinguishers in the model’s solution pool, allowing our tool to identify more valid DL distinguishers from a larger pool of higher quality candidates under limited computing resources. As an application, we find $12$-round and nine-round DL distinguishers for GIFT-64 and LELBC, respectively, and improve the best-known $13$-round DL distinguisher of PRESENT by one round. To the best of our knowledge, our nine-round DL distinguisher is the best distinguisher for LELBC in the single-key scenario.

$\textsf {Simon}$and$\textsf {Simeck}$are two famous lightweight block cipher families, both of which have good implementation performance benefiting from their extremely simple round functions. So, they are suitable and friendly in use for the Internet of Things devices that require high security but low-latency and low-energy. In this article, we aim to improve the mixed-integer linear programming/mixed-integer quadratic constraint programming (MILP/MIQCP)-based method, to find better differential-linear (DL) distinguishers for the above ciphers, which can be exploited to mount distinguishing or key-recovery attacks. In particular, first, we give the completely precise mixed-integer linear programming (MILP) model to describe the linear part, and utilize the general expressions of$\textsf {Gurobi}$optimizer to model middle part in a quite easy way. Second, to explore DL trails in a reasonable time, we propose two heuristic strategies to speed up the searching process. Lastly, we introduce the transforming technique, which exploits the clustering effect on DL trails, to improve the estimated correlation of the DL approximation. By applying our enhanced method, we improve the DL distinguisher correlation from$2^{-59.75}$to$2^{-59.62}$for 32-round$\textsf {Simon128}$, and extend the number of longest rounds of valid DL distinguishers for$\textsf {Simon32/48/64/96}$from$11/16/16/25$to$14/17/21/26$. For$\textsf {Simeck}$, we do not outperform the currently best work, but refresh Zhou et al.’s results (the first work to automate finding DL distinguishers for$\textsf {Simon/Simeck}$using MILP/MIQCP). Our work not only provides a new insight on the automatic DL cryptanalysis, but also further confirms that$\textsf {Simon}$and$\textsf {Simeck}$are sufficiently strong to resist the DL attacks.

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let q be a prime power. In this paper, by using the simplicial complexes of${\mathbb {F}}_{q}^{m}$with one single maximal element, we construct four families of linear codes over the ring${\mathbb {F}}_{q}+u{\mathbb {F}}_{q}$($u^{2}=0$), which generalizes the results of Wu et al. (2020). The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over${\mathbb {F}}_{q}$, including (near) Griesmer codes and distance-optimal codes. Moreover, it is shown that most of the Gray images are minimal or self-orthogonal codes which are useful in applications.

Twisted generalized Reed-Solomon (TGRS) codes are an extension of generalized Reed-Solomon (GRS) codes, and have recently attracted significant attention due to their potential for constructing non-GRS MDS codes. This paper presents an in-depth and comprehensive investigation of TGRS codes in their most general form, allowing arbitrary twists at arbitrary positions. First, we introduce a more precise definition of TGRS codes, namely (L,P)-TGRS codes, and provide a concise necessary and sufficient condition for them to be MDS, thereby generalizing previous results. Second, we explicitly characterize the parity check matrices of (L,P)-TGRS codes, and provide a sufficient condition for them to be self-dual. Finally, we investigate the non-GRS properties of (L,P)-TGRS codes via two approaches: the dimensions of Schur squares and combinatorial techniques. As a result, we obtain an infinite family of non-GRS MDS codes.

Frequency-hopping sequences (FHSs) with low Hamming correlation and wide gaps significantly contribute to the anti-interference performance in FH communication systems. This paper investigates FHSs with optimal Hamming correlation and controlled minimum gaps. We start with the discussion of the upper bounds on the minimum gaps of uniform FHSs and then propose a general construction of optimal uniform wide-gap FHSs with length 2land 3l, which includes the work by Li et al. in IEEE Trans. Inf. Theory, vol. 68, no. 1, 2022 as a special case. Furthermore, we present a recursive construction of FHSs with length 2l, which concatenate shorter sequences of known minimum gaps. It is shown that the resulting FHSs have the same Hamming correlation as the concatenation-ordering sequences. As applications, several known optimal FHSs are used to produce optimal FHSs with controlled minimum gaps.

We propose to study the construction of Boolean bent functions from cyclotomic mappings. By considering Dillon functions, Niho functions and Kasami functions as different branch functions respectively, we obtain three generic constructions from this new perspective. As a result, several infinite classes of bent functions belonging to the${\mathcal {PS}}_{ap}$class, class$\mathcal {H}$and the completed$\mathcal {MM}$class are derived, thereby providing simple representations of known classes of bent functions through cyclotomic mappings. In addition, computer experiments show that examples of bent functions outside these three well-known classes can also be obtained by selecting other branch functions.

The SM4 block cipher is standardized in ISO/IEC, and it is also the national standard of commercial cryptography in China. In this paper, we propose two new techniques called “split‐and‐join” and “off‐peak and stagger” to make SM4 more applicable to resource‐constrained environments. The area optimization method uses a 1‐bit data path while reducing the number of registers from 64 to 8 and the number of XOR gates from 194 to 8. As a result, we report a 1‐bit‐serial SM4 encryption circuit that occupies 1771 GE with a latency of 2,336 cycles. Additionally, the “off‐peak and stagger” technique compresses all the operations within the state update and key schedule into 32 clock cycles to reduce the latency. In other words, it takes 32 clock cycles to complete one round encryption. The new circuit occupies 1861 GE with a latency of 1,344 cycles. Moreover, we also discuss how to further reduce the latency by increasing the data path with a small area overhead to provide wider area‐latency tradeoffs for SM4. Our designs make SM4 competitive with many ciphers specifically designed for lightweight cryptography.

Dilithium is a signature scheme that is currently being standardized to the Module-Lattice-Based Digital Signature Standard by NIST. It is believed to be secure even against attacks from large-scale quantum computers based on lattice problems. The implementation efficiency is important for promoting the migration of current cryptography algorithms to post-quantum cryptography algorithms. In this article, we optimize the implementation of Dilithium with several new approaches proposed. Firstly, we improve the efficiency of parallel NTT implementations. The overhead of shuffling operations is reduced in our implementations, and fewer loading instructions are invoked for the precomputations. Then, we optimize the sampling and bit-packing of polynomial coefficients in Dilithium. We can handle double the number of coefficients within one register using a new approach for the sampling of secret key polynomials. The approaches proposed in this article are applicable to implementations under AVX2 and AVX-512 instruction sets. Take Dilithium2 as an illustration, our AVX2 implementation demonstrates improvements of 22.7%, 16.9%, and 13.5% for KeyGen, Sign, and Verify compared with the previous implementation.

In this paper, we construct a large family of projective linear codes over${\mathbb F}_{q}$from the general simplicial complexes of${\mathbb F}_{q}^{m}$via the defining-set construction, which generalizes the results of [IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weight distributions of this class of codes are completely determined. By using the Griesmer bound, we give a necessary and sufficient condition such that the codes are Griesmer codes and a sufficient condition such that the codes are distance-optimal. For a special case, we also present a necessary and sufficient condition for the codes to be near Griesmer codes. Moreover, by discussing the cases of simplicial complexes with one, two and three maximal elements respectively, the parameters and weight distributions of the codes are given more explicitly, which shows that the codes are at most 2-weight, 5-weight and 19-weight respectively. By studying the optimality of the codes for the three cases in detail, many infinite families of optimal linear codes with few weights over${\mathbb F}_{q}$are obtained, including Griesmer codes, near Griesmer codes and distance-optimal codes.

Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit relation between nonlinear complexities of finite-length binary sequences and their corresponding periodic sequences. Based on the relation, we propose two algorithms that can generate all periodic binary sequences with any prescribed nonlinear complexity.

A verifiable timed signature (VTS) scheme allows a signature to be time-locked to a known message for a predetermined duration denoted as$\mathsf {T}$. Verifiability ensures that anyone can verify that the time-lock contains a valid signature without completing the computation. In this paper, we introduce a novel VTS construction method based on the RSA group, designed to maintain a constant level of size. This approach serves as an improvement over the previous linear level size construction method (CCS 2020). First, we construct it by using a commitment to a valid RSA signature. This commitment can only be opened to a regular RSA signature after a sequential computation period. Our scheme utilizes a trapdoor verifiable delay function, RSA signatures, and a specialized zero-knowledge proof to instantiate the proposed scheme. We also conduct proofs in three aspects: correctness, soundness, and security. Furthermore, we identify potential applications for VTS and present a simple Bitcoin voting protocol based on an open vote protocol by utilizing VTS. Experimental results show that our scheme is more efficient compared to the construction of VTS (CCS 2020), reducing the signature size by at least 90.5% and lowering computational costs by at least 77%.

In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all binary sequences with length$n$and nonlinear complexity$c\geq n/2$, where$n$is an integer larger than 2. Furthermore, a formula is established to calculate the exact number of these sequences. The distribution of nonlinear complexity for these sequences is thus completely determined.

In this paper, we investigate the power function$F(x)=x^{d}$over the finite field$\mathbb {F}_{2^{4n}}$, where$n$is a positive integer and$d=2^{3n}+2^{2n}+2^{n}-1$. We prove that this power function is AP$c\text{N}$with respect to all$c\in \mathbb {F}_{2^{4n}}\setminus \{1\}$satisfying$c^{2^{2n}+1}=1$, and we determine its$c$-differential spectrum. To the best of our knowledge, this is the second class of AP$c\text{N}$power functions over finite fields of even characteristic. By the same proof ideas, we completely determine the differential spectrum of this function, and give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski.

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers$m$and$k$such that$\gcd (2^{m}+1,2^{k}+1)=1$, we investigate the locally-APN-ness property of the Niho type power function$F(x)=x^{s(2^{m}-1)+1}$over the finite field$\mathbb {F}_{2^{2m}}$for$s=(2^{k}+1)^{-1}$, where$(2^{k}+1)^{-1}$denotes the multiplicative inverse modulo$2^{m}+1$. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that$F(x)$is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for$2\leq m\leq 10$. In addition, we also determine the boomerang spectrum of$F(x)$by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications.

Extending previous results, we study a class of general quadrinomials over the field of size 22mwith odd m and characterize conditions under which they are permutations with 4-uniform BCT, a new and important parameter related to boomerang-style attacks. These permutations are known to have the best known nonlinearity. Numerical data also show that the inverse of these functions all have large algebraic degree, making them desirable for applications.

The expansion complexity is a new figure of merit for cryptographic sequences. In this paper, we present an explicit formula of the (irreducible) expansion complexity of ultimately periodic sequences over finite fields. We also provide improved upper and lower bounds on the$N$th irreducible expansion complexity when they are not explicitly determined. In addition, for some infinite sequences with given nonlinear complexity, a tighter upper bound of their$N$th expansion complexity is given.

In this paper, a class of p-ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the perfect nonlinear functions over Fp(m)and (m, s)-bent functions from F2(m)to F2(s), where p is an odd prime and m, s are positive integers. The weight distributions are completely determined by the sign of the Walsh transform of weakly regular bent functions and the size of the preimage of the employed (m, s)-bent functions at the zero point, respectively. As a special case, a class of optimal linear codes meeting Griesmer bound is obtained from our construction.

In Eurocrypt'18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over F2(2m)with m odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined.

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for$x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$being involutions over the finite field${\mathbb F}_{q}$, where$r\geq 1$and$s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form$x^{r}h(x^{s})$over${\mathbb F}_{q}$from given involutions over some subgroups of${\mathbb F}_{q}^{*}$by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form$x^{r}h(x^{s})$over${\mathbb F}_{q}$are obtained.

The nonlinear complexity of a periodic sequence s is the length of the shortest feedback shift register that can generate s, and its value is upper bounded by the least period of s minus 1. In this paper, a recursive approach that generates all periodic sequences with maximum nonlinear complexity is presented, and the total number of such sequences is determined. The randomness properties of these sequences are also examined.

Let p be a prime, n = 2m and d = 3pm- 2 with m ≥ 2, and gcd(d, pn- 1) = 1. In this paper, the correlation distribution between a p-ary m-sequence of period pn- 1 and its d-decimation sequence is investigated in a unified approach. Some results for the binary case are extended to the general case. It is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field Fpn. For an arbitrary odd prime p, it seems difficult to solve these two problems. However, for p = 3, by studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over F3n, the two problems are solved, and thus, the corresponding correlation distribution for d is completely determined. It is noteworthy that this is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.

In this paper, a family of new de Bruijn sequences is proposed through the construction of maximum-length nonlinear feedback shift registers (NFSRs). Let$k$be a positive integer and$p_{0}(x), p_{1}(x), \ldots , p_{k}(x)$be the primitive polynomials in$\mathbb {F}_{2}[x]$with their degrees strictly increasing and pairwise coprime. We determine the cycle structure and adjacency graphs of linear feedback shift registers (LFSRs) with characteristic polynomial$q(x)=\prod \nolimits _{i=0}^{k}p_{i}(x)$. In the case that$p_{0}(x)=1+x$, an algorithm is proposed to produce maximum-length NFSRs from these LFSRs, and it is shown that the algorithm can generate$O(2^{(2^{k}-1)n})~n$-stage maximum-length NFSRs with memory complexity$O(2^{k}kn)$and time complexity$O(2^{n-d_{k}}kn)$, where$n$and$d_{k}$are the degrees of$q(x)$and$p_{k}(x)$, respectively. Finally, we illustrate the proposed algorithm in the case of$k=2$. In this case, we prove that for any integer$n\geq 8$, the algorithm can produce$n$-stage maximum-length NFSRs with time complexity as low as$O(n^{{\rm {log}{log}}(n)}$).

In this paper, let n = 2k and d = 3 · 2k- 2 with k ≥ 3 and gcd(d, 2n- 1) = 1. Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary m-sequence of period 2n- 1 and its d-decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al.

In this paper, the cycle structure and adjacency graphs of a class of linear feedback shift registers (LFSRs) are determined. By recursively applying the D-morphism to the maximum-length LFSRs and representing the cycles by generating functions, a new family of maximum-length nonlinear feedback shift registers (NFSRs) are proposed based on the properties of these LFSRs. The number of NFSRs in the proposed family is also considered.

In this paper, a class of linear feedback shift registers (LFSRs) with characteristic polynomial (1 + x3)p(x) is discussed, where p(x) is a primitive polynomial of degree n > 2. The cycle structure and adjacency graphs of the LFSRs are determined. A new class of de Bruijn sequences is constructed from these LFSRs, and the number of de Bruijn sequences in the class is also considered. To illustrate the efficiency of constructing de Bruijn sequences from these LFSRs, an algorithm for producing some corresponding maximum-length nonlinear feedback shift registers with time and memory complexity O(n) is also proposed.

For an odd prime p and two positive integers m, k such that m/gcd (k,m) ≥ 3 is odd, let d be a positive integer satisfying d(pk+1)=2(mod pm-1). In this paper, the cross-correlation between a p-ary m-sequence and its d-decimated sequences is investigated, and the cross correlation distribution is completely determined. The relationship between the decimations d considered in this paper and some known ones is also studied. This paper generalizes some previous results, and also gives new decimations, which lead to low cross correlation.

In this paper, a new construction of zero-difference balanced functions defined on is given, where is an odd positive integer. Based on the generic constructions proposed by Ding, optimal constant composition codes and perfect difference systems of sets with new parameters can be generated from the zero-difference balanced functions constructed in this paper.

In this paper, a new generalized cyclotomy with respect to a positive odd integer is introduced, and a construction of frequency hopping sequence sets and two constructions of frequency hopping sequences are proposed as its applications. The frequency hopping sequence sets and frequency hopping sequences obtained in this paper can be optimal with respect to the Peng-Fan bound and Lempel-Greenberger bound, respectively. Further, the length of sequences in the optimal frequency hopping sequence sets can be any odd integer larger than 3. Some of them have new parameters.

This paper studies a class of binomial functions over the finite fields of odd characteristic and characterizes their bentness in terms of the Kloosterman sums. Numerical results show that the proposed class contains bent functions that are affinely inequivalent to all known monomial and binomial ones.

In this paper, we propose an interleaving construction of new sets of frequency hopping sequences from the known ones. By choosing suitable known optimal frequency hopping sequences and sets of frequency hopping sequences and then recursively applying the proposed construction, optimal frequency hopping sequences and sets of frequency hopping sequences with new parameters can be obtained.

Recently, new optimal Families${\cal S}$and${\cal U}$of quaternary sequences have been presented, and the optimal binary sequence Family${\cal V}$obtained from Family${\cal S}$under Gray map has been investigated as well. The two sequence Families${\cal U}$and${\cal V}$are optimal with respect to the well-known Sidelnikov bound and Welch bound, but their exact correlation distributions are not known until now. In this paper, their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of${\bf Z}_4$-valued quadratic forms.

In this paper, three constructions of balanced Boolean functions with optimal algebraic immunity are proposed. It is checked that, at least for small numbers of input variables, these functions have good behavior against fast algebraic attacks as well. Other cryptographic properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed. Lower bounds on the nonlinearity are proved, which are similar to the best bounds obtained for known Boolean functions resisting algebraic attacks and fast algebraic attacks. Moreover, it is checked that for the numbernof variables with 5 ≤n≤ 19, the proposedn-variable Boolean functions have in fact very good nonlinearity.

This paper follows the recent work of Helleseth, Kholosha, Johansen, and Ness to study the cross correlation between an m -sequence of period 2m- 1 and the d-decimation of an m-sequence of a shorter period 2n- 1 for an even number m = 2n. Assuming that d satisfies d(2l+ 1) = 2i(mod 2n- 1) for some l > 0 and i > 0, it is proved that the cross correlation takes on either exactly three or four values depending on whether I and n are coprime or not. The distribution of the cross-correlation values is also completely determined. Our results theoretically confirm the numerical data by Ness and Helleseth. It is conjectured that there are no other decimations that give at most four-valued cross correlation apart from the ones proved here.

In this paper, two new optimal families S and U of quadriphase sequences are presented. Compared to the family A constructed by Boztas and the family D investigated by Tang respectively, the proposed families have the same optimal correlation properties and family size, but larger linear spans.

In this paper, we construct two new families S and U of optimal quadriphase sequences. Compared to the family A constructed by Boztas et al and family D investigated by Tang et al respectively, the proposed families have the same optimal correlation properties and family size, but larger linear spans.

In this correspondence, new binary sequence families Fkof period 2n-1 are constructed for even n and any k with gcd(k,n)=2 if n/2 is odd or gcd(k,n)=1 if n/2 is even. The distribution of their correlation values is completely determined. These families have maximum correlation 2n/2+1and family size 23n/2+ 2n/2for odd n/2 or 23n/2+2n/2-1 for even n/2. The proposed families include the large set of Kasami sequences, where the k is taken as k=n/2+1.

A family of binary sequences is presented and proved to have optimal correlations and large linear span. It includes the small set of Kasami sequences, No sequence set and TN sequence set as special cases. A lower bound on the linear span of the family is provided. With suitable choices of parameters, it is proved that the family has exponentially larger linear spans than both No sequences and TN sequences.

A family of [2n- 1, 5n/2, 2n-1- 2n/2] codes with n even is proposed and the weight distribution is completely determined. These codes are applied to construct new families of binary sequences with period (2n$1), including the small set of Kasami sequences. The binary sequence families have the same family size and correlation values as the large set of Kasami sequences

Keyboards, mice, and joy sticks are the most popular controlling and navigation devices in current 3D game. However, they are quite unnatural. In this paper, we propose a novel scheme to estimate the user's head pose and the estimation result is used for navigating in game. The novel scheme based on Markov model fusing appearance, color and motion information. The experimental results demonstrate that the proposed approach which is used for 3D game controlling, is real-time and robust, and can give players more immersiveness.