Linear Codes Sample Clauses

Linear Codes. A linear q-ary code of length n and rank k is a subspace C with dimension k of the vector space Fn. The vectors in C are called codewords. The size of a code is the number of codewords, and is thus equal to qk. The weight of a word w Fn is the number of non-zero components and the distance between two words is the Hamming distance between them (equivalently, the weight of their difference). The minimal distance d of a linear code C is the minimum weight of its non-zero codewords, or equivalently, the minimum distance between any two distinct codewords. A code for an alphabet of size q, of length n, rank k, and minimal distance d is called an (n, k, d)q-code. Such a code can be used to detect up to d − 1 errors (because if a codeword is sent and fewer than d − 1 errors occur, it will not get transformed to another codeword), and correct up to [(d− 1)/2♩ errors (because for any received word, there is a unique codeword within distance [(d − 1)/2♩). For linear codes, encoding of a (row vector) word W ∈ Fk is performed by an algorithm C.Encode : Fk → Fn, which is the multiplication of W by a so-called
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Linear Codes. A linear q-ary code of length n and rank k is a subspace C with dimension k of the vector space Fn. The vectors in C are called codewords. The size of a code is the number of codewords it contains, and is thus equal to qk. The weight of a word w Fn is the number of its non-zero components, and the distance between two words is the Hamming distance between them (equivalently, the weight of their difference). The minimal distance d of a linear code C is the minimum weight of its non-zero codewords, or equivalently, the minimum distance between any two distinct codewords. A code for an alphabet of size q, of length n, rank k, and minimal distance d is called an (n, k, d)q-code. Such a code can be used to detect up to d − 1 errors (because if a codeword is sent and fewer than d− 1 errors occur, it will not get transformed to another codeword), and correct up to |(d − 1)/2∫ errors (because for any received word, there is a unique codeword within distance |(d − 1)/2∫). For linear codes, the encoding of a (row vector) word W ∈ Fk is performed by an algorithm C.Encode : Fk → Fn, which is the multiplication of W by a so-called “generating matrix” G ∈ Fk×n (which defines an injective linear map). This leads to a row-vector codeword W G =: c C Fn. The ▇▇▇▇▇▇▇▇▇ bound states that for any linear code, it holds that k + d n + 1. A maximum distance separable (or MDS) code satisfies k + d = n + 1. Since d = n k + 1, MDS codes are fully described by the parameters (q, n, k). Such an (n, k)q-MDS code can correct up to (n k)/2 errors; it can detect if there are errors whenever there are no more than n k of them. For a matrix G generating an MDS code, any set of k columns of G are linearly independent. For a thorough introduction to linear codes and proofs of all statements in this short overview we refer the reader to [Rot06]. Observe that a linear code, due to the linearity of its encoding algorithm, is not a primitive designed to hide anything about the encoded message (e.g., a popular choice for the generating matrix is G := (Ik ) with Ik being the k k identity matrix). However, we show in the following lemma how to turn an MDS code into a RSS scheme with additional smoothness guarantees.
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Related to Linear Codes

  • Architecture The Private Improvements shall have architectural features, detailing, and design elements in accordance with the Project Schematic Drawings. All accessory screening walls or fences, if necessary, shall use similar primary material, color, and detailing as on the Private Improvements.

  • Network Interconnection Architecture Each Party will plan, design, construct and maintain the facilities within their respective systems as are necessary and proper for the provision of traffic covered by this Agreement. These facilities include but are not limited to, a sufficient number of trunks to the point of interconnection with the tandem company, and sufficient interoffice and interexchange facilities and trunks between its own central offices to adequately handle traffic between all central offices within the service areas at a P.01 grade of service or better. The provisioning and engineering of such services and facilities will comply with generally accepted industry methods and practices, and will observe the rules and regulations of the lawfully established tariffs applicable to the services provided.

  • Programming Processor is not responsible for programming or reprogramming of fuel dispensers.

  • Interfaces GTE provides the CLECs with choices for access to OSS pre-ordering, ordering, maintenance and repair systems. Availability of the interfaces is fundamental to the CLEC being able to effectively do business with GTE. Additionally, in many instances, CLEC personnel must work with the service personnel of GTE. Measurements in this category assess the availability to the CLECs of systems and personnel at GTE work centers.

  • BRAND NAME OR EQUALS/DEVIATIONS Unless otherwise specified, the mention of a particular manufacturer’s brand name or number in the specifications does not imply that this particular good is the only one that will be considered for purchase. This reference is intended solely to designate the type or quality of good that will be acceptable. Equal offers will be considered and must include descriptive literature and/or specifications. Failure to provide descriptive literature and/or specifications with equal offers will result in the disqualification of the bid. The determination as to whether any alternate good or service is or is not equal shall be made solely by the County and such determination shall be final and binding upon all bidders. The County reserves the right to request and review additional information to make such a determination. Although the County provides for the consideration of alternate bids, it reserves the right to make an award in the best interest of the County. Award may not necessarily be given to the lowest bid offered. The Bidder shall be responsible for reading very carefully, and understanding completely, the requirements and the specifications of the items bid upon. Unless the bid is in response to a “Brand Name or Equal” requirement, deviations from the specifications will only be considered if requested in writing prior to the date and time specified for receipt of bids. Deviations, if accepted, will be specifically addressed in writing via an addendum to this Invitation for Bids. Any goods or services that are not in compliance with the specifications will not be accepted.