Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube F2m. In this paper, we study punctured Reed-Muller codes Pr(m, b), whose positions are restricted to the m-tuples of a given Hamming weight b. In combinatorial terms, this paper concerns m-variate Boolean polynomials of any degree r, which are evaluated on a Hamming sphere of some radius b in F2m. Codes Pr(m, b) inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical b-layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes Pr(m, b) and find their distances for the arbitrary values of parameters r, m, and b. Finally, we describe recursive (successive cancellation) decoding of these codes.

Deep neural networks (DNNs) have recently become the state of the art technology in speech recognition systems. In this pa-per we propose a new approach to constructing large high qual-ity unsupervised sets to train DNN models for large vocabulary speech recognition. The core of our technique consists of two steps. We first redecode speech logged by our production rec-ognizer with a very accurate (and hence too slow for real-time usage) set of speech models to improve the quality of ground truth transcripts used for training alignments. Using confidence scores, transcript length and transcript flattening heuristics de-signed to cull salient utterances from three decades of speech per language, we then carefully select training data sets consist-ing of up to 15K hours of speech to be used to train acoustic models without any reliance on manual transcription. We show that this approach yields models with approximately 18K con-text dependent states that achieve 10 % relative improvement in large vocabulary dictation and voice-search systems for Brazil-ian Portuguese, French, Italian and Russian languages. Index Terms: large unsupervised training sets, data selection,

Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube Fm2. In this paper, we study punctured Reed-Muller codes Pr(m, b) whose positions form a spherical b-layer and include all m-tuples of a given Hamming weight b. These punctured codes inherit some recursive properties of the original RM codes and can be built from the shorter codes, by decomposing a spherical b-layer into sub-layers of smaller dimensions. However, codes Pr(m, b) cannot be formed by the recursive Plotkin construction. We analyze recursive properties of these codes and find their code distances for arbitrary values of parameters r, m, and b.

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F2mand let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials Kbm(w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.

Consider a binary Reed-Muller code RM(r, m) defined on the full set of binary m-tuples and let this code be punctured to the spherical layer S(b) that includes only m-tuples of a given Hamming weight b. More generally, we can consider punctured RM codes RM(r, m, B) restricted to some set B of several spherical layers S(b), b ϵ B. In this paper we specify this construction for the biorthogonal codes RM(1, m) and the Hadamard codes H(m). It is shown that the overall weight of any code vector in a punctured code H(m, B) is determined by the weight w of its information block. More specifically, this weight depends only on the values of the Krawtchouk polynomials Kbm(w) for all b ϵ B. We further refine our codes by limiting the possible weights w of the input information blocks. As a result, we obtain sequences of codes that meet or closely approach the Griesmer bound.