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As a particular subclass of cyclic codes, BCH codes have wide applications in storage devices, communication systems, consumer electronics and other fields. However, parameters of BCH codes are unknown in general. In this paper, we investigate parameters of BCH codes of length q m + 1 λ where λ ∣ q + 1 .Some new techniques are employed to study the coset leaders. For any odd prime power q and m = 4 , 8 , or m ≥ 12 and m ≡ 4 (mod 8) , the second, the third and the fourth largest coset leaders modulo q m + 1 are determined, and the dimensions of some BCH codes of length q m + 1 with large designed distances are given. For 1 < λ < q + 1 , the first few largest coset leaders and the coset leaders modulo q m + 1 λ in the range 1 to q ⌊ (m + 1) / 2 ⌋ λ are studied, and the dimensions of some BCH codes of length q m + 1 λ are given as well. The BCH codes presented in this paper are LCD codes and have a sharper lower bound on the minimum distance than the well-known BCH bound. [ABSTRACT FROM AUTHOR]

Telecommand (TC) plays a crucial role in the success of high-altitude balloon experiments. Bose, Ray-Chaudhuri, Hocquenghem (BCH) codes are commonly employed to ensure reliable command operation. The Balloon Facility (BF) of Tata Institute of Fundamental Research (TIFR) uses a TC system based on BCH (31,16) coding technique, to control balloon and payload operations. This paper presents prototyping and implementation of TC encoder and decoder using Spartan 6 Field Programmable Gate Array (FPGA). The code is written in Very high-speed integrated circuit Hardware Description Language (VHDL). Simulation and synthesis are done using Xilinx ISE 14.7 design suite. Simulation results show the design is robust. The TC encoder is implemented in a commercial FPGA development board and the TC decoder is implemented in a specially designed FPGA board, successfully. This paper presents the salient features of the TC system in use and the implementation of the system using FPGA. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m a positive integer. Suppose that a ≥ 2 is a factor of q m − 1 such that m is the multiplicative order of q modulo n : = q m − 1 a. Firstly, for m odd and m even, we present some necessary and sufficient conditions on dual-containing non-primitive BCH codes of length n with the maximum designed distances over F q , respectively. The results show that the maximum designed distances of dual-containing BCH codes in this paper are larger than those in [2, Theorem 3]. Secondly, we explicitly determine the dimensions of some dual-containing non-primitive BCH codes of length n , and finally, we construct some new quantum codes with relatively large minimum distances. [ABSTRACT FROM AUTHOR]