Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel

Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R is below 1-2p, for stabilizer codes: we also derive an improved upper bound of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant - low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings.Comment: 32 page

Shadow bounds for self-dual codes

Conway and Sloane (1990) have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a self-dual binary code of length n is at most 4[n/24]+4, except when n mod 24=22, when the bound is 4[n/24]+6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory)

On the parameters of codes for the Lee and modular distance

AbstractWe introduce the concept of a weakly metric association scheme, a generalization of metric schemes. We undertake a combinatorial study of the parameters of codes in these schemes, along the lines of [9]. Applications are codes over Zq for the Lee distance and arithmetic codes for the modular distance.Our main result is an inequality which generalizes both the Delsarte upper bound on covering radius, and the MacWilliams lower bound on the external distance, yielding a strong necessary existence condition on completely regular codes.The external distance (in the Lee metric) of some self-dual codes of moderate length over Z5 is computed

Shadow bounds for self-dual codes

Conway and Sloane (1990) have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a self-dual binary code of length n is at most 4[n/24]+4, except when n mod 24=22, when the bound is 4[n/24]+6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory)

Codes over rings : maximum distance separability and self-duality /

Una parte imporante de la teoría de códigos es la de determinar cotas del número de palabras de un código. Uno de los problemas fundamentales de la teoría de códigos es encontrar códigos con la máxima distancia mínima d. Los investigadores han encontrado diferentes cotas superiores e inferiores para los códigos lineales y no lineales, por ejemplo cotas de Plotkin, Johnson, Singleton, Elias, Linear Programming, Griesmer, Gilbert y Varshamov. En esta tesis se ha estudiado la cota de Singleton, que es una cota superior de la distancia mínima de un código, y se han definido los códigos Z2Z4-aditivos a distancia máxima separable (MDS). Dos cotas diferentes se presentan en este trabajo en el que se han caracterizado todos los códigos Z2Z4-aditivos a distancia máxima separable con respecto a la cota de Singleton (MDSS) y condiciones en los parámetros para códigos Z2Z4-aditivos a distancia máxima separable con respecto a la cota obtenida a partir del rango (MDSR). La generación de nuevos códigos ha sido siempre un tema interesante, dando lugar al estudio de las propiedades de estos nuevos códigos generados y a establecer nuevos resultados. Los códigos autoduales son una clase importante de códigos. Hay numerosas construcciones de códigos autoduales a partir de objetos combinatorios. En este trabajo se han dado dos métodos para generar códigos autoduales a partir de esquemas de asociación de clase 3; las construcciones pure y bordered. Con estos dos métodos, se han obtenido códigos binarios autoduales a partir de esquemas de asociación de clase 3 no simétricos y códigos sobre Zk a partir de esquemas de asociación rectangulares. Borges, Dougherty y Fernández-Córdoba en 2011 presentaron un método para generar nuevos códigos Z2Z4-aditivos autoduales a partir de otros códigos Z2Z4-aditivos autoduales extendiendo su longitud. En este trabajo se ha comprobado si las propiedades como separabilidad, antipodalidad y el tipo del código se mantienen o no cuando se utiliza este método.Bounds on the size of a code are an important part of coding theory. One of the fundamental problems in coding theory is to find a code with largest possible distance d. Researchers have found different upper and lower bounds on the size of linear and nonlinear codes e.g., Plotkin, Johnson, Singleton, Elias, Linear Programming, Griesmer, Gilbert and Varshamov bounds. In this dissertation we have studied the Singleton bound, which is an upper bound on the minimum distance of a code, and have defined maximum distance separable (MDS) Z2Z4 additive codes. Two different forms of these bounds are presented in this work where we have characterized all maximum distance separable Z2Z4-additive codes with respect to the Singleton bound (MDSS) and strong conditions are given for maximum distance separable Z2Z4-additive codes with respect to the rank bound (MDSR). Generation of new codes has always been an interesting topic, where one can study the properties of these newly generated codes and establish new results. Self-dual codes are an important class of codes. There are numerous constructions of self-dual codes from combinatorial objects. In this work we have given two methods for generating self-dual codes from 3-class association schemes, namely pure construction and bordered construction. Binary self-dual codes are generated by using these two methods from non-symmetric 3-class association schemes and self-dual codes from rectangular association schemes are generated over Zk. Borges, Dougherty and Fernández-Córdoba in 2011 presented a method to generate new Z2Z4-additive self-dual codes from the existing Z2Z4-additive selfdual codes by extending their length. In this work we have verified whether properties like separability, antipodality and code Type are retained or not, when using this method

Determining when the absolute state complexity of a Hermitian code achieves its DLP bound

Let g be the genus of the Hermitian function field H/F(q)2 and let C-L(D,mQ(infinity)) be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of C-L(D,mQ(infinity)). Here we determine when this lower bound is tight and when it is not. For m less than or equal to n-2/2 or m greater than or equal to n-2/2 + 2g, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of C-L(D,mQ(infinity)), which improves the DLP lower bound. Next we give a good coordinate order for C-L(D,mQ(infinity)). With this good order, the state complexity of C-L(D,mQ(infinity)) achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of C-L(D,mQ(infinity)) (for those values of m for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases. A straightforward application of these results is that if C-L(D,mQ(infinity)) is self-dual, then its state complexity (with respect to the lexicographic coordinate order) achieves its DLP bound of n /2 - q(2)/4, and, in particular, so does its absolute state complexity

Linear programming bounds for doubly-even self-dual codes

Using a variant of linear programming method we derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives d/n <=166315 + o(1), thus improving on the Mallows– Odlyzko–Sloane bound of 1/6. To establish this, we prove that in any doubly even-self-dual code the distance distribution is asymptotically upper-bounded by the corresponding normalized binomial distribution in a certain interval

An improved upper bound on the minimum distance of doubly-even self-dual codes

We derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives limn→∞ sup d/n <=(5-5^3/4)/10 <0.165630, thus improving on the Mallows-Odlyzko-Sloane bound of 1/6 and our recent bound of 0.16631