We introduced error-correcting codes and linear codes in the last lecture. In this lecture we will discuss in some more details properties of linear codes and we’ll describe classical examples of linear codes. We will also show the Hamming bound, which is a bound that relates the distance, rate, and block length parameters of codes and is tight for the Hamming codes. Recall that a (n, k, d)q co...

The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum M...

A rational (resp. integral) distance set is a subset S of the plane R such that for all s, t ∈ S, the distance between s and t is a rational number (resp. is an integer). Huff [4] considered rational distance sets S of the following form: given distinct a, b ∈ Q∗, S contains the four points (0,±a) and (0,±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q∗. Such a point (x, 0) m...

Given a polygonal path P with vertices p1, p2, . . . , pn ∈ Rd and a real number t ≥ 1, a path Q = (pi1 , pi2 , . . . , pik) is a t-distance-preserving approximation of P if 1 = i1 < i2 < . . . < ik = n and each straight-line edge (pij , pij+1) of Q approximates the distance between pij and pij+1 along the path P within a factor of t. We present exact and approximation algorithms that compute s...

These problems are inspired by a careful study of the papers of concerning bipartite distance-regular graphs. Throughout these notes we let Γ = (X, R) denote a bipartite distance-regular graph with diameter D ≥ 3 and standard module V = C X. We fix a vertex x ∈ X and let E denote the corresponding dual primitive idempotents. We define the matrices R = D i=0 E * i+1 AE * i , L= D i=0 E * i−1 AE ...

a manufacturing process cannot be released to production until it has been proven to be stable. also, we cannot begin to talk about process capability until we have demonstrated stability in our process. this means that the process variation is the result of random causes only and all assignable or special causes have been removed. in complicated manufacturing processes, such as drilling proces...

Let Fq denote a finite field with q elements and let V = (Fq)m,n be the Fq-vector space of matrices over Fq of type (m,n). On V we define the so-called rank metric distance by d(A,B) = rank(A−B) for A,B ∈ V . Clearly, the distance d is a translation invariant metric on V . A subset C ⊆ V endowed with the metric d is called a rank metric code with minimum distance d(C) = min {d(A,B) | A 6= B ∈ V...

Let $$D\subset {\mathbb {C}}^n$$ be a bounded domain. A pair of distinct boundary points $$\{p,q\}$$ D has the visibility property provided there exist compact subset $$K_{p,q}\subset D$$ and open neighborhoods $$U_p$$ p $$U_q$$ q, such that real geodesics for Kobayashi metric which join in intersect $$K_{p,q}$$ . Every Gromov hyperbolic convex domain enjoys any couple points. The Goldilocks do...

Let denote a Q-polynomial distance-regular graph with diameter d¿ 3. We show that if the valency is at least three, then the intersection number p 12 is at least two; consequently the girth is at most six. We then consider a condition on the dual eigenvalues of that must hold if is the quotient of an antipodal distance-regular graph of diameter D¿ 7; we call a pseudoquotient whenever this condi...

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy ai = 0 for 0 ≤ i ≤ D − 1 and aD 6= 0. We show that Γ is a polygon, a folded cube, or an Odd graph.