The Hasse derivative image of a repeated-root cyclic code is typically non-cyclic. This prevents a direct ideal-theoretic calculation of derivative-twisted intersections and first leads to cyclic containers and two-sided bounds. This paper shows that, after expanding codewords in Hasse–Taylor coordinates at the roots of the underlying polynomial, the higher-order Hasse derivative operator decomposes into independent local blocks and its image becomes a coordinate subspace. This yields exact closed-form dimensions, expressed entirely in terms of the generator multiplicities and the base-prime digits of the derivative order, for the derivative image, for the derivative-twisted intersection of two codes and its self-intersection specialization, and for the smallest cyclic code containing the image together with its non-cyclic defect. As a consequence, for every positive derivative order in the admissible range, the derivative image is cyclic only when it is zero, and the multiplicity-drop and Lucas-refined containers developed here, as well as the rank of the derivative operator, are recovered as immediate relaxations or special cases. Applying the intersection-pair construction to a code paired with its derivative image produces entanglement-assisted quantum error-correcting codes whose dimension and entanglement cost are exact functions of the multiplicity digits and the derivative order, so that the derivative order tunes the entanglement consumption. For the minimum distance, a punctured matrix-product decomposition gives an exact formula for the derivative image in terms of the minimum distances of a finite family of simple-root constituents of length n0. Combining this formula with the corresponding constituent formula for the dual gives the complete distance parameter of the resulting EAQECCs without enumerating the non-cyclic derivative images.
The Hasse derivative image of a repeated-root cyclic code is typically non-cyclic. This prevents a direct ideal-theoretic calculation of derivative-twisted intersections and first leads to cyclic containers and two-sided bounds. This paper shows that, after expanding codewords in Hasse–Taylor coordinates at the roots of the underlying polynomial, the higher-order Hasse derivative operator decomposes into independent local blocks and its image becomes a coordinate subspace. This yields exact closed-form dimensions, expressed entirely in terms of the generator multiplicities and the base-prime digits of the derivative order, for the derivative image, for the derivative-twisted intersection of two codes and its self-intersection specialization, and for the smallest cyclic code containing the image together with its non-cyclic defect. As a consequence, for every positive derivative order in the admissible range, the derivative image is cyclic only when it is zero, and the multiplicity-drop and Lucas-refined containers developed here, as well as the rank of the derivative operator, are recovered as immediate relaxations or special cases. Applying the intersection-pair construction to a code paired with its derivative image produces entanglement-assisted quantum error-correcting codes whose dimension and entanglement cost are exact functions of the multiplicity digits and the derivative order, so that the derivative order tunes the entanglement consumption. For the minimum distance, a punctured matrix-product decomposition gives an exact formula for the derivative image in terms of the minimum distances of a finite family of simple-root constituents of length n0. Combining this formula with the corresponding constituent formula for the dual gives the complete distance parameter of the resulting EAQECCs without enumerating the non-cyclic derivative images.
This thesis investigates the algebraic structure, construction, encoding, and decoding of quasi-cyclic (QC) and quasi-twisted (QT) codes and their applications to quantum error correction. For selected repeated-root and simple-root QC codes, generator-matrix-based construction methods are developed using the Generalized Discrete Fourier Transform (GDFT) and Discrete Fourier Transform (DFT), respectively. These methods enable efficient encoding by matrix-vector multiplication and provide computational advantages over codeword-based constructions. Spectral decoding procedures are also studied, including a GDFT-based folding framework that relates decoding of repeated-root QC codes to that of associated simple-root codes. Furthermore, a correspondence between certain repeated-root QC codes over finite fields and simple-root QT codes over finite chain rings is established. Building on these classical structures, one-generator QT codes of index 3 are investigated for quantum-code construction. Their Hermitian duals are characterized, and sufficient conditions for Hermitian self-orthogonality are derived, yielding stabilizer quantum error-correcting codes with good and, in several cases, optimal parameters. Structured syndrome-based decoding is developed by relating stabilizer measurements to classical decoding of the underlying QT codes. An extension technique preserving Hermitian self-orthogonality is introduced, producing additional stabilizer codes and corresponding decoding procedures. Finally, entanglement-assisted quantum error-correcting codes are constructed from 1-generator QT codes through Hermitian rank conditions, yielding maximal-entanglement codes with competitive parameters. Overall, the thesis establishes QC and QT codes as a unified algebraic framework for efficient classical coding, stabilizer quantum error correction, and entanglement-assisted quantum coding.
In this paper, we study the hulls of $ \mathbb{Z}_p\mathbb{Z}_p[v] $- cyclic codes for any prime $ p $, where $ v^2 = v $. Firstly, we derive the generator polynomials of the hulls of separable cyclic codes (SCCs) over $ \mathbb{Z}_p\mathbb{Z}_p[v] $ and determine their dimensions using the concept of a generating function in combinatorics. We also enumerate the SCCs over $ \mathbb{Z}_p\mathbb{Z}_p[v] $ with hulls of a fixed dimension. Next, we present the generator polynomials of the hulls of non-separable cyclic codes (NSCCs) over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ with coprime odd block lengths and determine their dimensions. Additionally, we enumerate the NSCCs over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ with hulls of a fixed dimension. As an application, we use the hulls of NSCCs over $ \mathbb{Z}_2\mathbb{Z}_2[v] $ to construct some good entanglement-assisted quantum error-correcting codes (EAQECCs).
Let C be an [n,k,d] linear code over $\F_{q^2}$ with parity-check matrix H. The rank $r=\Rank(HH^{\dagger})$ measures how far the Hermitian dual C⊥h is from being Hermitian self-orthogonal, and it is the least number of coordinates that must be adjoined to C⊥h to make it Hermitian self-orthogonal; equivalently, the Hermitian hull C∩C⊥h has dimension (n−k)−r. In this paper, we develop a structure theory of the Hermitian hull of a cyclic code and apply it to the construction of quantum stabilizer codes. For a cyclic code with defining set T⊆Zn, we show that r is additive over the orbits of the involution τ(ν)=−qν on the q2-cyclotomic cosets and satisfies a parity constraint. A parameter dictionary exhibits r as a gauge trading one-for-one against the code length, and a complementation duality shows that the complement of T realises the logical-qubit count of the quantum code. The extreme values of r are settled by closed subset-sum criteria, and the τ-invariant defining sets are shown to be the unique locus on which the Hermitian dual is again cyclic. On the distance side, we determine the Hermitian dual distance of an r-fold extension of an arbitrary linear code exactly, and we prove that the admissible extension vectors lie in the image of HH†, which reduces the search space from q2(n−k) to q2r. Applied to cyclic, quasi-cyclic and BCH codes, the construction improves 23 best-known parameters in Grassl's code tables, one of them provably optimal.
New QEC codes and EAQEC codes from repeated-root cyclic codes of length 2rps
Let p be an odd prime and r,s,m be positive integers. In this study, we initiate our exploration by delving into the intricate structure of all repeated-root cyclic codes and their duals with a length of 2rps over the finite field Fpm. Through the utilization of CSS and Steane's constructions, a series of new quantum error-correcting (QEC) codes are constructed with parameters distinct from all previous constructions. Furthermore, we provide all maximum distance separable (MDS) cyclic codes of length 2rps, which are further utilized in the construction of QEC MDS codes. Finally, we introduce a significant number of novel entanglement-assisted quantum error-correcting (EAQEC) codes derived from these repeated-root cyclic codes. Notably, these newly constructed codes exhibit parameters distinct from those of previously known constructions.
The σ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes
Let SLAut(Fqn) denote the group of all semilinear isometries on Fqn, where q=pe is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the σ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be σ dual-containing and σ self-orthogonal. We prove that dimFq(Hullσ(C))=dimFq(Hullσ(C⊥σ)). We prove that for any integer h with max{0,k1−k2}≤h≤dimFq(C1∩C2⊥σ), there exists a linear code C2,h monomially equivalent to C2 such that dimFq(C1∩C2,h⊥σ)=h, where Ci is an [n,ki]q linear code for i=1,2. We show that given an [n,k,d]q linear code C, there exists a monomially equivalent [n,k,d]q linear code Ch, whose σ dual code has minimum distance d′, such that there exist an [[n,k−h,d;n−k−h]]q EAQECC and an [[n,n−k−h,d′;k−h]]q EAQECC for every integer h with 0≤h≤dimFq(Hullσ(C)). Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to σ hulls.
In this work, we study (r,t,x) quantum locally recoverable codes (qLRCs) with locality r, t recovery sets per qudit, and intersection parameter x. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an (r,t,x)-qLRC if and only if the underlying classical codes are (r,t,x) classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing (r,t,x)-cLRCs, which yield binary (r,t,x)-qLRCs via the CSS construction. For CSS (r,t,x)-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.