On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets

In this work, we study quantum error-correcting codes obtained by using Steane-enlargement. We apply this technique to certain codes defined from Cartesian products previously considered by Galindo et al. in [4]. We give bounds on the dimension increase obtained via enlargement, and additionally give an algorithm to compute the true increase. A number of examples of codes are provided, and their parameters are compared to relevant codes in the literature, which shows that the parameters of the enlarged codes are advantageous. Furthermore, comparison with the Gilbert-Varshamov bound for stabilizer quantum codes shows that several of the enlarged codes match or exceed the parameters promised by the bound.Comment: 12 page

Steane-Enlargement of Quantum Codes from the Hermitian Curve

In this paper, we study the construction of quantum codes by applying Steane-enlargement to codes from the Hermitian curve. We cover Steane-enlargement of both usual one-point Hermitian codes and of order bound improved Hermitian codes. In particular, the paper contains two constructions of quantum codes whose parameters are described by explicit formulae, and we show that these codes compare favourably to existing, comparable constructions in the literature.Comment: 11 page

On one-round reliable message transmission

In this paper, we consider one-round protocols for reliable message transmission (RMT) when $t$ out of $n=2t+1$ available channels are controlled by an adversary. We show impossibility of constructing such a protocol that achieves a transmission rate of less than $\Theta(n)$ for constant-size messages and arbitrary reliability parameter. In addition, we show how to improve two existing protocols for RMT to allow for either larger messages or reduced field sizes

On nested code pairs from the Hermitian curve

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum codes [Ketkar et al. 2006]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance.Comment: 28 page

Private Randomness Agreement and its Application in Quantum Key Distribution Networks

We define a variation on the well-known problem of private message transmission. This new problem called private randomness agreement (PRA) gives two participants access to a public, authenticated channel alongside the main channels, and the 'message' is not fixed a priori. Instead, the participants aim to agree on a random string completely unknown to a computationally unbounded adversary. We define privacy and reliability, and show that PRA cannot be solved in a single round. We then show that it can be solved in three rounds, albeit with exponential cost, and give an efficient four-round protocol based on polynomial evaluation.Comment: 6 page

Private Product Computation Using Quantum Entanglement

In this article, we show that a pair of entangled qubits can be used to compute a product privately. More precisely, two participants with a private input from a finite field can perform local operations on a shared, Bell-like quantum state, and when these qubits are later sent to a third participant, the third participant can determine the product of the inputs, but without learning more about the individual inputs. We give a concrete way to realize this product computation for arbitrary finite fields of prime order.</p