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...ODES 2541 for any , we have that . Since is a monotonically increasing function of , we have that for typical random linear codes. The weight distribution of typical random linear codes is studied in =-=[16]-=-. Let with be the number of codewords of weight for typical random linear codes. It is shown that , which is a monotonically increasing function of for . Thus, we have that for . From the above result...

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...ibution of the coset leaders represents the number of correctable errors for each weight. Considering this fact, the error correction capabilities for certain specific codes are completely determined =-=[1]-=-–[4]. For general linear codes, some bounds on the number of correctable errors were presented in [5]–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of corre...

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...rial set for . Since every larger half is an uncorrectable error, we have the relation A codeword is called minimal if for implies . Basic properties and applications of minimal codewords are seen in =-=[12]-=-. Let denote the set of minimal codewords in . Then we have the following property [6, Corollary 5]: is a trial set for (4) then is also a trial set for (5) It follows from the above fact that the set...

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... this fact, the error correction capabilities for certain specific codes are completely determined [1]–[4]. For general linear codes, some bounds on the number of correctable errors were presented in =-=[5]-=-–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of correctable errors for them was known only for weight [8]. Determining the number of correctable errors of...

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...ucture is useful for error analysis since the correctable (and uncorrectable) errors are characterized by the maximal correctable (and minimal uncorrectable) errors. Helleseth, Kløve, and Levenshtein =-=[6]-=- analyzed an asymptotic error correctability of binary linear codes using the monotone error structure, and introduced two useful notions, larger half and trial set, which characterizes the minimal un...

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...e number of correctable errors were presented in [5]–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of correctable errors for them was known only for weight =-=[8]-=-. Determining the number of correctable errors of weight for the first-order Reed–Muller codes is equivalent to determining the number of Boolean functions with nonlinearity , and the nonlinearity of ...

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... an important criterion for cryptographic system, block ciphers and stream ciphers. Many studies have been conducted on the nonlinearity of Boolean functions in cryptography. For further details, see =-=[17]-=-, [9] and references therein. For an integer is defined recursively as . for for Proof: First we observe that . Let and be codewords in . Then where denotes the concatenation of and , and . Since all ...

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...ion of the coset leaders represents the number of correctable errors for each weight. Considering this fact, the error correction capabilities for certain specific codes are completely determined [1]–=-=[4]-=-. For general linear codes, some bounds on the number of correctable errors were presented in [5]–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of correctab...

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...Consider and with . The monotone structure is the following property: If is a correctable error, then is also correctable. If is uncorrectable, then is also uncorrectable. Using this structure, Zémor =-=[11]-=-, [18] elucidated that the error probability after the maximum likelihood decoding over the binary symmetric channels displays a threshold behavior. Helleseth et al. [6] studied this structure and int...

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...er and with . The monotone structure is the following property: If is a correctable error, then is also correctable. If is uncorrectable, then is also uncorrectable. Using this structure, Zémor [11], =-=[18]-=- elucidated that the error probability after the maximum likelihood decoding over the binary symmetric channels displays a threshold behavior. Helleseth et al. [6] studied this structure and introduce...