Quaternary Plotkin Constructions and Quaternary Reed-Muller Codes
New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of
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Department of Information and Communications Engineering, Universitat Aut` onoma de Barcelona, 08193-Bellaterra, Spain 2 Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia
Abstract. New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4 -linear codes have the same parameters as the codes in the classical binary linear Reed-Muller family. Keywords: Quaternary codes, Plotkin constructions, Reed-Muller codes, Z4-linear codes.
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Introduction
In [13] Nechaev introduced the concept of Z4 -linearity of binary codes and later Hammons, Kumar, Calderbank, Sloane and Sol´e, see [7], showed that several families of binary codes are Z4 -linear. In [7] it is proved that the binary linear Reed-Muller code RM (r, m) is Z4 -linear for r = 0, 1, 2, m − 1, m and is not Z4 -linear for r = m − 2 (m ≥ 5). In a subsequent work, Hou, Lahtonen and Koponen, [8] proved that RM (r, m) is not Z4 -linear for 3 ≤ r ≤ m − 2. In [7] the construction of Reed Muller codes, QRM(r, m), based on Z4 linear codes is introduced such that after doing modulo two we obtain the usual binary linear Reed-Muller (RM ) codes. In [2,3] such family of codes is studied and their parameters are computed as well as the dimension of the kernel and rank. In [15] some kind of Plotkin construction was used to build a family of additive ReedMuller codes and also in [17] the Plotkin construction was utilized to obtain a sequence of quaternary linear Reed-Muller like codes. In both last quoted constructions, images of the obtained codes under the Gray map are binary codes with the same parameters as the classical binary linear RM codes. Moreover, on the other hand, in [9,10] all the non-equivalent Z4 -linear extended 1-perfect codes and their duals, the Z4 -linear Hadamard codes, are classified. It is a natural question to ask if there exist families of quaternary linear
This work has been partially supported by the Spanish MEC and the European FEDER Grant MTM2006-03250 and also by the UAB grant PNL2006-13.
S. Bozta¸s and H.F. Lu (Eds.): AAECC 2007, LNCS 4851, pp. 148–157, 2007. c Springer-Verlag Berlin Heidelberg 2007
Quaternary Plotkin Constructions and Quaternary Reed-Muller Codes
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codes such that, after the Gray map, the corresponding Z4 -linear codes have the same parameters as the well known family of binary linear RM codes. In this new families, like in the usual RM (r, m) family, the code with parameters (r, m) = (1, m) should be a Hadamard code and the code with parameters (r, m) = (m − 2, m) should be an extended 1-perfect code. It is well known that an easy way to built the RM family of codes is by using the Plotkin construction (see [12]). So, it seems a good matter of study to try to generalize the Plotkin construc
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