Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of [14], several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in [14] are upper bounded by the code alphabet...

The Main Conjecture on maximum distance separable (MDS) codes states that, except for some special cases, the maximum length of a q-ary linear MDS code of is q+1. This conjecture does not hold true for near maximum distance separable codes because of the existence of q-ary near-MDS elliptic codes having length bigger than q+1. An interesting related question is whether a near-MDS elliptic code ...

For q ∈ (0, 1), let Bq denote the limit q-Bernstein operator. In this paper, the distance between Bq and Br for distinct q and r in the operator norm on C[0, 1] is estimated, and it is proved that 1 6 ‖Bq −Br‖ 6 2, where both of the equalities can be attained. To elaborate more, the distance depends on whether or not r and q are rational powers of each other. For example, if rj 6= qm for all j,...

Suppose that we are given sample access to an unknown distribution p over n elements and an explicit distribution q over the same n elements. We would like to reject the null hypothesis “p = q” after seeing as few samples as possible, when p 6= q, while we never want to reject the null, when p = q. Well-known results show that Θ( √ n/ 2) samples are necessary and sufficient for distinguishing w...

Article history: Received 14 December 2007 Revised 9 September 2008 Accepted 11 September 2008 Available online 25 September 2008 Communicated by Naoki Saito We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal q-quasinorm is also the sparsest one. This generalizes, and slightly improves, a similar result for the ...

Recently Perlmutter et al. (1999) suggested a positive value of Einstein’s cosmological constant Λ on the basis of luminosity distances from type-Ia supernovae (the “SN-method”). However, Λ world models had earlier been proposed by Hoell & Priester (1991) and Liebscher et al. (1992a,b) on the basis of quasar absorption-line data (the “Q-method”). Employing more general repulsive fluids (“dark e...

Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting information from decoherence and noise. In this work, we construct six families of new EAQECCs lengths $$n=(q^2+1)/a$$ , $$n=q^2+1$$ $$n=(q^2+1)/2$$ cyclic codes, where $$a=m^2+1$$ ( $$m\ge 1$$ is odd) q an odd prime power with the form $$a|(q+m)$$ or $$a|(q-m)$$ . Moreover, those are entangleme...

Let denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, . . . , θD is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence θi−1 − βθi + θi+1 = 0 (0 < i < D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have conjectured that q is real if the diameter D is suffi...

Let Σ : M → 2 \ {∅} be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y, ‖·‖). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a singlevalued function g from a given closed convex set V ⊂ C(M, Y ). In an abstract setting this problem is posed as the extremal problem supt∈M ...

The degree distance of a graph which is a degree analogue of the Wiener index of the graph. Let G = TUV C6[2p, q] be the carbon nanotubes covered by C6, formulas for calculating the degree distance of armchair polyhex nanotubes TUV C6[2p, q] are provided. Mathematics Subject Classification: 05C05, 05C12