Web(b) Sphere-packing bound (9). (c) Minimum-distance bound (11). (d) Minimum-distance bound (12). (e) The new bound (14).R denotes the critical rate. Each of the curves (d)–(e) includes a segment of the common tangent to the curve and the sphere-packing exponent. method in coding theory, discovered recently by the authors in [3] and [25]. In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible … See more An original message and an encoded version are both composed in an alphabet of q letters. Each code word contains n letters. The original message (of length m) is shorter than n letters. The message is converted into an n … See more Main article: Delone set For an $${\displaystyle A_{q}(n,d)}$$ code C (a subset of $${\displaystyle {\mathcal {A}}_{q}^{n}}$$), the covering radius of C is the smallest … See more • Griesmer bound • Singleton bound • Gilbert-Varshamov bound • Plotkin bound See more Preliminary definitions An alphabet set $${\displaystyle {\mathcal {A}}_{q}}$$ is a set of symbols with $${\displaystyle q}$$ elements. The set of strings of length See more It follows from the definition of $${\displaystyle d}$$ that if at most $${\displaystyle t=\left\lfloor {\frac {1}{2}}(d-1)\right\rfloor }$$ See more Codes that attain the Hamming bound are called perfect codes. Examples include codes that have only one codeword, and codes that are the whole of A perfect code may … See more
Packing properties of binary mixtures in disordered sphere …
Webnew bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres. 2010 Mathematics Subject Classification: 52C17, 90C22 (primary) 1. Introduction How densely can one pack given objects into a given container? Problems of this WebThe densest binary sphere packings in the α-x plane of small to large sphere radius ratio α and small sphere relative concentration x have historically been very difficult to determine. Previous research had led to the prediction that these packings were composed of a few known "alloy" phases including, for example, the AlB 2 (hexagonal ω ... increased costs and schedule slips
Connecting Packing Efficiency of Binary Hard Sphere Systems to …
WebSep 30, 2024 · For SECCs, we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases. We also obtain improved … WebSphere packing This table gives the best packing densities known for congruent spheres in Euclidean spaces of dimensions 1 through 48 and 56, 64, and 72, along with the best … WebThe binary Golay codeG23has lengthn= 23, dimension 12;and distance 7. (a) Prove thatG23meets the sphere packing bound with equality. (b) Suppose that we perform the nearest neighbor decoding ofG23, i.e., given a vectory 2F23 2;find c 2 G23that satisfiesd(c;y) d(c′;y) for allc′2 G23. Prove that this codewordcequalsy+x, wherex increased costs 意味