Linear codes with complementary duals (LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Nonbinary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over F _q (q > 3) is equivalent to a Euclidean LCD code and any linear code over F _q ² (q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over F _q with q > 3 exists if there is an [n, k, d]-linear code over F _q and an [n, k, d]-linear Hermitian LCD code over F _q ² with q > 2 exists if there is an [n, k, d]-linear code over F _q ² . Hence, when q > 3 (resp. q > 2) q-ary Euclidean (resp. q ² -ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q ² -ary linear codes). This gives a direct proof that every triple of parameters [n, k, d] which is attainable by linear codes over F _q with q > 3 (resp. over F _q ² with q > 2) is attainable by Euclidean LCD codes (resp. by Hermitian LCD codes). In particular there exist families of q-ary Euclidean LCD codes (q > 3) and q ² -ary Hermitian LCD codes (q > 2) exceeding the asymptotical Gilbert-Varshamov bound. Further, we give a second proof of these results using the theory of Gröbner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results on the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.

In order to protect the privacy of individual data, meantime guaranteeing the utility of big data, the privacy preserving data aggregation is widely researched, which is a feasible solution since it not only preserves the statistical feature of the original data, but also masks single user’s data. With smart meter owning the capability of connecting to Internet, the aggregation area extends to the virtual area rather than a traditional physical area. However, in a virtual aggregation area, the users’ membership maybe frequently changes, if while executing the aggregation protocol for the traditional area, the overhead is not ignorable. In this paper, the homomorphic encryption and ID-based signature are employed to design a dynamic membership data aggregation (DMDA) scheme, which reduces the complexity on a new user’s joining and an old user’s quitting. In addition, the operation center obtains the sum of the data in the virtual aggregation area, meantime knows nothing about single user’s data. Comparing with traditional privacy-preserving data aggregation scheme, DMDA is more suitable for next-generation smart grid and other Internet of Things environments.

An [n,k,n-k+1] linear code is called an MDS code. An [n,k,n-k] linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11,6,5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system S(5,6,12) with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for t ≥ 2 remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over GF(3s) holding an infinite family of 3-designs and an infinite family of near MDS codes over GF(22s) holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal.

Linear complementary dual (LCD) codes are a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called sidechannel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular algebraic geometry LCD codes which could be good candidates to be resistant against SCA. We firstly provide a construction scheme for obtaining LCD codes from any algebraic curve. Then, some explicit LCD codes from elliptic curves are presented. Maximum distance separable (MDS) codes are of the most importance in coding theory due to their theoretical significance and practical interests. In this paper, all the constructed LCD codes from elliptic curves are MDS or almost MDS. Some infinite classes of LCD codes from elliptic curves are optimal due to the Griesmer bound. Finally, we also derive some explicit LCD codes from hyperelliptic curves and Hermitian curves.