Homological Product Codes

Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most $\sqrt{n}$ qubits, where $n$ is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight $n^a$ for any $a>0$. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.Comment: 49 page

Decoding and constructions of codes in rank and Hamming metric

As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions. We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x). An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes. Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles. We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin

A new family of optical code sequences for spectral-amplitude-coding optical CDMA systems

A new code structure for spectral-amplitude-coding optical code-division multiple-access system based on double-weight (DW) code families is proposed. The DW code has a fixed weight of two. By using a mapping technique, codes that have a larger number of weights can be developed. Modified double-weight (MDW) code is a DW code family variation that has variable weights of greater than two. The newly proposed code possesses ideal cross-correlation properties and exists for every natural number n. Based on theoretical analysis and simulation, MDW code is shown here to provide a much better performance compared to Hadamard and modified frequency-hopping codes

A new code for optical code division multiple access systems

A new code structure based on Double-Weight (DW) code families is proposed for Spectral-Amplitude-Coding Optical Code Division Multiple Access (OCDMA) system. The constraint of a constant weight of 2 for the DW code can be relaxed using a mapping technique. By using this technique, codes that have a larger number of weight can be developed. Modified Double-Weight (MDW) Code is another variation of a DW code family that can has a variable weight greater than two. The MDW code possesses ideal cross-correlation properties and exists for every natural number n. A much better performance can be provided by using the MDW code compared to the existing codes such as Hadamard and Modified Frequency-Hopping (MFH) codes. This has been demonstrated from the theoretical analysis and simulation