Quantum error-correcting codes associated with graphs

One-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes

We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...

Constacyclic Codes over Group Ring (Zq[v])/G

Recently, codes over some special finite rings especially chain rings have been studied. More recently, codes over finite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over fields. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum ...

Hypermap-Homology Quantum Codes (Ph.D. thesis)

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev’s toric code, have been discussed widely...

Good Quantum Error-Correcting Codes based on Sparse Graphs

Asymmetric Quantum Codes on Toric Surfaces

Asymmetric quantum error-correcting codes are quantum codes defined over biased quantum channels: qubitflip and phase-shift errors may have equal or different probabilities. The code construction is the Calderbank-ShorSteane construction based on two linear codes. We present families of toric surfaces, toric codes and associated asymmetric quantum error-correcting codes.