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... decade due to many new and diverse applications of these codes. Some examples are constrained and partial-response channels [4], interleaving schemes [15], orthogonal frequency-division multiplexing =-=[16]-=-, multidimensional burst-error-correction [17], and error-correction for flash memories [18]. The increased interest is also due to new attempts to settle the existence question of perfect codes in th...

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..., ∀k. Golay proposed a construction for complementary sequence pairs with sequence elements taken from alphabet {1, −1} [1,2,4,3]. This was extended by Turyn and others [5,6], and generalised to sets =-=[7,8,9,10]-=-, to other alphabets [11,12,13] to arrays [14,15,16,9,17,18,19,20], to near-complementarity [15,21], and to complete complementary codes [22]. A typical requirement is that any pair so constructed com...

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...iently large number of distinct sequences from Construction 2. Also, we observed that the code rate Rdist for a given m is higher than the ones in any other known near-complementary sequences [15][16]=-=[20]-=-. Remark 1: When t = 0, we realize that the near-complementary sequences from Construction 2 are equivalent to the codewords in Golay complementary sets of size 4 and PMEPR ≤ 4 from Construction 14 in...

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...which are known as complementary pairs, form a special case of CSS. Complementary pairs with binary (i.e. 71) elements were first studied in [5] and are usually referred to as Golay pairs (GP). CSS have been applied to radar pulse compression [7], multiple-input–multiple-output (MIMO) radars [8], ultrasonic ranging [9], synthetic aperture imaging [10], and ultrasonography [11]. In addition to active sensing systems, CSS have applications in code-division multipleaccess (CDMA) communication schemes [12], ultra wideband (UWB) communications [13], orthogonal frequencydivision multiplexing (OFDM) [14,15], channel estimation [16], and data hiding [17]. Due to such a wide range of applications, the construction of CSS has been an active area of research during the last decades. The majority of research results on CSS have been concerned with the analytical construction of GP or CSS for restricted sequence lengths N. For example, it is shown in [18] that GPs exist for lengths of the form N 2a10b26g where a,b and g are non-negative integers. Some conditions on the existence of CSS can be found in [19] and [20]. Furthermore, Ref. [20] considers the extension of GP to general CSS. A theoretical as...

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... of length n with high rate, large minimum distance, and PMEPR significantly lower than n. This problem is considered to be difficult, and only a few explicit constructions are known [22], [8], [21], =-=[23]-=-. An even more challenging problem is the construction of code sequences with nonvanishing rate and increasing length n such that the PMEPR grows strictly less than with O(n). Some results on the exis...

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... . In Z4 it generates the same lattice as the generator matrix G(2, 2) = 1 1 1 1 0 2 0 2 0 0 2 2 0 0 0 4 . Reducing the entries of G(2, 2) into zeroes and ones yields H2 = 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 . Clearly, the rows of G(m, j) are linearly independent. Let Λ(m, j) be the lattice whose generator matrix is G(m, j), and C(m, j) the code reduced from Λ(m, j), over Z2j , whose generator matrix is G(m, j). C(m, j) was constructed by a completely different approach for the control of the peak-tomean envelope power ratio in orthogonal frequency-division multiplexing in [21], where its size and minimum distance were discussed. The following lemma is an immediate result from the recursive construction of Hm. Lemma 2: The number of rows with Hamming weight 2i, 0 ≤ i ≤ m, in Hm is ( m i ) . By Lemma 2 and by the definition of G(m, j), for each `, j ≤ ` ≤ m, there exist rows in G(m, j) with Manhattan weight 2`. These are the only weights of rows in G(m, j). Theorem 5: • The minimum Manhattan distance of Λ(m, j) is 2j . • The volume of the lattice Λ(m, j) is Πji=02 (j−i)(mi ). • Λ(m, j) is reduced to the code C(m, j). C(m, j) has minimum Lee distance 2j . In Section I...

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...ond-order cosets of generalized first-order Reed-Muller codes RM2h(1, n). As they are members of RM2h(1, n), the sequences have good pairwise distinguishability. The work was subsequently extended by =-=[22, 26, 27]-=- and by numerous other authors. [19, 20, 17, 10, 15, 21] show that the complementary set construction is primarily an array construction, where sequence sets are obtained by considering suitable proje...

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.... This way the encoding and decoding procedures become simpler, in particular for larger block lengths. For the details about encoding and decoding of the proposed code classes we refer the reader to =-=[9]-=-. VI. CONCLUSION In this paper a large family of sequences lying in complementary sets have been presented. Moreover the classical Reed–Muller codes have been generalized in a novel manner. We have sh...