Abstract
A \(q\)-ary code of length \(n\) is termed an equitable symbol weight code, if each symbol appears among the coordinates of every codeword either \(\lfloor n/q \rfloor \) or \(\lceil n/q \rceil \) times. This class of codes was proposed recently by Chee et al. in order to more precisely capture a code’s performance against permanent narrowband noise in power line communication. In this paper, two series of new equitable symbol weight codes of optimal sizes meeting the Plotkin bound are constructed via combinatorial designs.
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Acknowledgments
Research supported by the National Natural Science Foundation of China under Grant No. 11271280.
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Communicated by V. A. Zinoviev.
Appendix: The required starter–adder pairs in the Proof of Lemma 3.3
Appendix: The required starter–adder pairs in the Proof of Lemma 3.3
\(u\) | \(S\) | \(A\) | \(u\) | \(S\) | \(A\) | \(u\) | \(S\) | \(A\) |
|---|---|---|---|---|---|---|---|---|
\(\{ 1_3, 4_4, 2_0, 3_3, \infty _0\}\) | \(0_0\) | \(\{ 1_2, 2_4, 3_0, 5_1,\infty _0 \}\) | \(2_0\) | \(\{ 2_0, 1_2, 3_3, 6_0, \infty _0\}\) | \(2_0\) | |||
\(\{ 2_3, 1_4, 0_0, 4_0, \infty _1\}\) | \(2_0\) | \(\{ 0_4, 3_2, 4_2, 5_2,\infty _1 \}\) | \(4_0\) | \(\{ 1_4, 3_4, 4_4, 5_1, \infty _1\}\) | \(0_0\) | |||
\(\{ 1_0, 0_3, 3_2, 4_2, \infty _2\}\) | \(3_0\) | \(\{ 3_4, 1_3, 4_0, 0_0,\infty _2 \}\) | \(1_0\) | \(\{ 2_2, 5_4, 4_3, 6_3, \infty _2\}\) | \(4_0\) | |||
5 | \(\{ 1_1, 3_4, 2_1, 0_1, \infty _3\}\) | \(1_0\) | 6 | \(\{ 1_1, 2_3, 5_4, 0_3,\infty _3 \}\) | \(3_0\) | 7 | \(\{ 2_1, 5_0, 0_3, 3_1, \infty _3\}\) | \(6_0\) |
\(\{ 2_2, 3_1, 4_3, 0_2, \infty _4\}\) | \(4_0\) | \(\{ 2_2, 3_1, 4_1, 0_1,\infty _4 \}\) | \(5_0\) | \(\{ 1_1, 4_2, 6_2, 0_2, \infty _4\}\) | \(5_0\) | |||
\(\{ 1_2, 3_0, 2_4, 4_1, 0_4 \}\) | \(R\) | \(\{ 2_0, 1_4, 3_3, 5_0, 4_3 \}\) | \(0_0\) | \(\{ 2_4, 4_0, 1_3, 5_2, 0_1, \}\) | \(1_0\) | |||
\(\{ 2_ 2, 3_ 4, 7_ 3, 0_ 0, \infty _0\}\) | \(6_0\) | \(\{ 2_1, 4_4, 1_0, 5_3, 0_2 \}\) | \(R\) | \(\{ 1_0, 6_4, 2_3, 3_2, 0_4 \}\) | \(3_0\) | |||
\(\{ 3_ 3, 6_ 3, 7_ 4, 9_ 3, \infty _1\}\) | \(8_0\) | \(\{ 4_ 1, 5_ 4, 6_ 3, 8_ 3, \infty _0\}\) | \(7_0\) | \(\{ 3_0, 0_0, 5_3, 4_1, 6_1 \}\) | \(R\) | |||
\(\{ 1_ 4, 5_ 3, 7_ 1, 8_ 1, \infty _2\}\) | \(4_0\) | \(\{ 1_ 1, 2_ 3, 4_ 0, 7_ 1, \infty _1\}\) | \(3_0\) | \(\{ 2_ 3, 3_ 4, 5_ 1, 6_ 1, \infty _0\}\) | \(4_0\) | |||
\(\{ 1_ 3, 4_ 4, 5_ 1, 9_ 4, \infty _3\}\) | \(3_0\) | \(\{ 2_ 4, 5_ 2, 7_ 0, 0_ 3, \infty _2\}\) | \(1_0\) | \(\{ 1_ 3, 6_ 4, 7_ 0, 0_ 3, \infty _1\}\) | \(3_0\) | |||
\(\{ 2_ 3, 7_ 2, 8_ 2, 0_ 2, \infty _4\}\) | \(9_0\) | \(\{ 3_ 3, 4_ 3, 5_ 3, 0_ 2, \infty _3\}\) | \(2_0\) | \(\{ 1_ 1, 2_ 0, 5_ 4, 0_ 4, \infty _2\}\) | \(5_0\) | |||
10 | \(\{ 1_ 2, 3_ 0, 4_ 3, 8_ 0, 9_ 0 \}\) | \(5_0\) | 9 | \(\{ 1_ 3, 3_ 2, 8_ 4, 0_ 0, \infty _4\}\) | \(4_0\) | 8 | \(\{ 2_ 1, 4_ 1, 6_ 0, 7_ 4, \infty _3\}\) | \(0_0\) |
\(\{ 2_ 1, 4_ 1, 5_ 2, 6_ 0, 9_ 2 \}\) | \(7_0\) | \(\{ 2_ 1, 3_ 0, 7_ 4, 8_ 2, 0_ 4 \}\) | \(8_0\) | \(\{ 3_ 2, 4_ 3, 6_ 3, 0_ 1, \infty _4\}\) | \(7_0\) | |||
\(\{ 4_ 2, 5_ 0, 6_ 1, 7_ 0, 8_ 4 \}\) | \(0_0\) | \(\{ 3_ 1, 6_ 4, 7_ 3, 8_ 0, 0_ 1 \}\) | \(5_0\) | \(\{ 1_ 4, 2_ 2, 3_ 0, 4_ 4, 7_ 1 \}\) | \(2_0\) | |||
\(\{ 1_ 1, 2_ 0, 3_ 1, 6_ 4, 0_ 4 \}\) | \(2_0\) | \(\{ 1_ 0, 2_ 0, 5_ 1, 6_ 0, 8_ 1 \}\) | \(0_0\) | \(\{ 2_ 4, 3_ 1, 4_ 2, 5_ 0, 0_ 2 \}\) | \(1_0\) | |||
\(\{ 1_ 0, 2_ 4, 4_ 0, 6_ 2, 0_ 3 \}\) | \(1_0\) | \(\{ 1_ 4, 2_ 2, 4_ 2, 6_ 1, 7_ 2 \}\) | \(6_0\) | \(\{ 1_ 2, 3_ 3, 5_ 3, 6_ 2, 7_ 2 \}\) | \(6_0\) | |||
\(\{ 3_ 2, 5_ 4, 8_ 3, 9_ 1, 0_ 1 \}\) | \(R\) | \(\{ 1_ 2, 3_ 4, 4_ 4, 5_ 0, 6_ 2 \}\) | \(R\) | \(\{ 1_ 0, 4_ 0, 5_ 2, 7_ 3, 0_ 0 \}\) | \(R\) | |||
\(\{ 1_ 1, 2_ 2, 8_ 1, 0_ 1, \infty _0\}\) | \(8_0\) | \(\{ 4_ 3, 5_ 4, 7_ 1, 11_ 4, \infty _0\}\) | \(3_0\) | \(\{ 2_ 1, 7_ 4, 11_ 3, 0_ 1, \infty _0\}\) | \(8_0\) | |||
\(\{ 3_ 1, 4_ 1, 5_ 4, 10_ 4, \infty _1\}\) | \(4_0\) | \(\{ 6_ 0, 8_ 1, 9_ 4, 10_ 1, \infty _1\}\) | \(9_0\) | \(\{ 5_ 1, 9_ 1, 10_ 2, 11_ 0, \infty _1\}\) | \(7_0\) | |||
\(\{ 3_ 0, 6_ 0, 8_ 2, 10_ 1, \infty _2\}\) | \(9_0\) | \(\{ 6_ 1, 9_ 2, 10_ 3, 0_ 1, \infty _2\}\) | \(7_0\) | \(\{ 2_ 2, 6_ 1, 8_ 0, 0_ 0, \infty _2\}\) | \(12_0\) | |||
\(\{ 1_ 2, 4_ 3, 6_ 3, 8_ 0, \infty _3\}\) | \(6_0\) | \(\{ 2_ 3, 3_ 4, 4_ 2, 0_ 2, \infty _3\}\) | \(6_0\) | \(\{ 3_ 1, 7_ 1, 11_ 2, 12_ 3, \infty _3\}\) | \(5_0\) | |||
\(\{ 3_ 2, 4_ 4, 9_ 1, 0_ 0, \infty _4\}\) | \(3_0\) | \(\{ 1_ 4, 4_ 1, 10_ 4, 11_ 1, \infty _4\}\) | \(10_0\) | \(\{ 1_ 2, 5_ 2, 10_ 0, 0_ 3, \infty _4\}\) | \(9_0\) | |||
11 | \(\{ 1_ 3, 4_ 2, 5_ 0, 7_ 0, 9_ 0 \}\) | \(1_0\) | 12 | \(\{ 3_ 0, 6_ 2, 8_ 3, 10_ 0, 0_ 0 \}\) | \(1_0\) | 13 | \(\{ 1_ 4, 6_ 3, 8_ 4, 9_ 4, 12_ 0 \}\) | \(10_0\) |
\(\{ 6_ 4, 7_ 2, 8_ 3, 9_ 3, 10_ 2 \}\) | \(7_0\) | \(\{ 1_ 3, 2_ 2, 4_ 0, 9_ 1, 11_ 3 \}\) | \(2_0\) | \(\{ 2_ 4, 5_ 4, 7_ 2, 9_ 0, 12_ 2 \}\) | \(2_0\) | |||
\(\{ 2_ 3, 6_ 2, 7_ 1, 10_ 0, 0_ 2 \}\) | \(2_0\) | \(\{ 2_ 0, 5_ 1, 6_ 4, 7_ 0, 11_ 0 \}\) | \(11_0\) | \(\{ 1_ 0, 6_ 2, 8_ 2, 9_ 3, 0_ 2 \}\) | \(1_0\) | |||
\(\{ 2_ 0, 3_ 4, 5_ 2, 10_ 3, 0_ 3 \}\) | \(10_0\) | \(\{ 1_ 2, 3_ 2, 5_ 3, 8_ 4, 0_ 3 \}\) | \(5_0\) | \(\{ 3_ 4, 4_ 1, 5_ 0, 6_ 0, 9_ 2 \}\) | \(6_0\) | |||
\(\{ 3_ 3, 4_ 0, 5_ 1, 7_ 3, 9_ 4 \}\) | \(0_0\) | \(\{ 1_ 0, 5_ 0, 7_ 4, 10_ 2, 11_ 2 \}\) | \(4_0\) | \(\{ 3_ 2, 4_ 4, 5_ 3, 8_ 1, 11_ 4 \}\) | \(3_0\) | |||
\(\{ 1_ 0, 2_ 4, 5_ 3, 8_ 4, 0_ 4 \}\) | \(5_0\) | \(\{ 2_ 1, 3_ 1, 7_ 3, 8_ 0, 9_ 0 \}\) | \(0_0\) | \(\{ 4_ 3, 6_ 4, 7_ 0, 10_ 4, 0_ 4 \}\) | \(11_0\) | |||
\(\{ 1_ 4, 2_ 1, 6_ 1, 7_ 4, 9_ 2 \}\) | \(R\) | \(\{ 2_ 4, 3_ 3, 6_ 3, 7_ 2, 8_ 2 \}\) | \(8_0\) | \(\{ 2_ 3, 3_ 0, 4_ 0, 10_ 1, 12_ 4 \}\) | \(0_0\) | |||
\(\{ 1_ 1, 2_ 0, 3_ 0, 13_ 3, \infty _0\}\) | \(2_0\) | \(\{ 1_ 1, 4_ 4, 5_ 2, 9_ 3, 0_ 4 \}\) | \(R\) | \(\{ 1_ 3, 2_ 0, 3_ 3, 4_ 2, 12_ 1 \}\) | \(4_0\) | |||
\(\{ 2_ 2, 3_ 2, 10_ 2, 0_ 1, \infty _1\}\) | \(9_0\) | \(\{ 3_ 1, 4_ 2, 5_ 3, 7_ 0, \infty _0\}\) | \(13_0\) | \(\{ 1_ 1, 7_ 3, 8_ 3, 10_ 3, 11_ 1 \}\) | \(R\) | |||
\(\{ 5_ 0, 13_ 1, 15_ 3, 0_ 4, \infty _2\}\) | \(6_0\) | \(\{ 2_ 0, 7_ 3, 1_ 0, 3_ 4, \infty _1\}\) | \(5_0\) | \(\{4_ 1, 9_ 2, 10_ 2, 11_ 0, \infty _0\}\) | \(6_0\) | |||
\(\{ 3_ 4, 6_ 1, 9_ 3, 10_ 0, \infty _3\}\) | \(4_0\) | \(\{11_ 0, 0_ 4, 2_ 2, 14_ 2, \infty _2\}\) | \(11_0\) | \(\{1_ 4, 2_ 3, 6_ 1, 10_ 1, \infty _1\}\) | \(7_0\) | |||
\(\{ 6_ 2, 8_ 4, 9_ 0, 10_ 4, \infty _4\}\) | \(15_0\) | \(\{10_ 0, 4_ 4, 3_ 0, 5_ 1, \infty _3\}\) | \(6_0\) | \(\{0_ 0, 13_ 4, 9_ 3, 11_ 2, \infty _2\}\) | \(12_0\) | |||
\(\{ 5_ 2, 11_ 0, 12_ 1, 13_ 0, 0_ 3 \}\) | \(10_0\) | \(\{10_ 1, 9_ 2, 7_ 1, 1_ 1, \infty _4\}\) | \(10_0\) | \(\{2_ 4, 4_ 2, 9_ 1, 10_ 4, \infty _3\}\) | \(11_0\) | |||
\(\{ 1_ 0, 2_ 3, 8_ 1, 10_ 3, 14_ 1 \}\) | \(0_0\) | \(\{ 9_ 0, 13_ 3, 12_ 1, 3_ 2, 2_ 1 \}\) | \(1_0\) | \(\{1_ 3, 5_ 0, 6_ 3, 13_ 2, \infty _4\}\) | \(10_0\) | |||
\(\{ 1_ 2, 4_ 4, 11_ 3, 12_ 2, 0_ 2 \}\) | \(13_0\) | \(\{ 2_ 3, 13_ 1, 5_ 2, 6_ 0, 11_ 4 \}\) | \(7_0\) | \(\{5_ 1, 2_ 0, 12_ 1, 7_ 0, 8_ 0 \}\) | \(1_0\) | |||
\(\{ 1_ 4, 5_ 3, 7_ 4, 8_ 2, 11_ 4 \}\) | \(1_0\) | \(\{ 8_ 3, 10_ 2, 6_ 2, 13_ 0, 12_ 0 \}\) | \(12_0\) | \(\{6_ 4, 11_ 4, 1_ 1, 9_ 4, 7_ 1 \}\) | \(9_0\) | |||
16 | \(\{ 2_ 4, 5_ 4, 7_ 3, 9_ 2, 14_ 0 \}\) | \(12_0\) | 15 | \(\{ 1_ 4, 13_ 4, 8_ 2, 6_ 4, 7_ 4 \}\) | \(0_0\) | 14 | \(\{4_ 3, 7_ 2, 12_ 4, 3_ 1, 0_ 4 \}\) | \(5_0\) |
\(\{ 6_ 0, 10_ 1, 12_ 4, 14_ 2, 15_ 4 \}\) | \(14_0\) | \(\{ 9_ 3, 12_ 2, 6_ 3, 14_ 4, 10_ 3 \}\) | \(2_0\) | \(\{13_ 0, 1_ 2, 6_ 0, 0_ 1, 3_ 2 \}\) | \(4_0\) | |||
\(\{ 2_ 1, 8_ 3, 9_ 1, 13_ 4, 15_ 2 \}\) | \(5_0\) | \(\{13_ 2, 0_ 2, 1_ 3, 8_ 0, 4_ 0 \}\) | \(4_0\) | \(\{8_ 3, 2_ 2, 11_ 3, 3_ 4, 0_ 2 \}\) | \(2_0\) | |||
\(\{ 4_ 0, 8_ 0, 11_ 1, 13_ 2, 15_ 0 \}\) | \(7_0\) | \(\{ 2_ 4, 7_ 2, 0_ 0, 8_ 1, 11_ 3 \}\) | \(14_0\) | \(\{2_ 1, 8_ 4, 12_ 3, 1_ 0, 4_ 4 \}\) | \(3_0\) | |||
\(\{ 3_ 3, 4_ 3, 6_ 4, 9_ 4, 14_ 3 \}\) | \(3_0\) | \(\{14_ 3, 11_ 2, 3_ 3, 8_ 4, 0_ 1 \}\) | \(3_0\) | \(\{4_ 0, 3_ 0, 5_ 4, 0_ 3, 12_ 2 \}\) | \(8_0\) | |||
\(\{ 1_ 3, 4_ 1, 6_ 3, 7_ 0, 0_ 0 \}\) | \(11_0\) | \(\{ 5_ 4, 12_ 4, 10_ 4, 14_ 0, 0_ 3 \}\) | \(9_0\) | \(\{7_ 3, 8_ 2, 5_ 2, 3_ 3, 13_ 3 \}\) | \(13_0\) | |||
\(\{ 3_ 1, 5_ 1, 7_ 1, 11_ 2, 12_ 3 \}\) | \(8_0\) | \(\{11_ 1, 4_ 3, 9_ 1, 1_ 2, 5_ 0 \}\) | \(8_0\) | \(\{5_ 3, 8_ 1, 10_ 3, 11_ 1, 12_ 0 \}\) | \(0_0\) | |||
\(\{ 4_ 2, 7_ 2, 12_ 0, 14_ 4, 15_ 1 \}\) | \(R\) | \(\{ 9_ 4, 14_ 1, 12_ 3, 4_ 1, 6_ 1 \}\) | \(R\) | \(\{7_ 4, 6_ 2, 10_ 0, 9_ 0, 13_ 1 \}\) | \(R\) |
\(u\) | \(S\) | \(A\) | \(S\) | \(A\) | \(S\) | \(A\) |
|---|---|---|---|---|---|---|
\(\{ 3_ 3, 13_ 3, 0_ 4, 15_ 1, \infty _0\}\) | \(9_0\) | \(\{ 3_ 0, 5_ 2, 10_ 4, 2_ 3, 11_ 2 \}\) | \(2_0\) | \(\{15_ 0, 4_ 3, 6_ 0, 0_ 1, 1_ 2 \}\) | \(3_0\) | |
\(\{ 6_ 3, 11_ 0, 13_ 2, 9_ 0, \infty _1\}\) | \(5_0\) | \(\{ 1_ 1, 16_ 1, 2_ 4, 11_ 3, 4_ 0 \}\) | \(7_0\) | \(\{14_ 4, 13_ 4, 5_ 3, 11_ 4, 15_ 3 \}\) | \(14_0\) | |
17 | \(\{10_ 0, 9_ 1, 3_ 4, 8_ 4, \infty _2\}\) | \(13_0\) | \(\{11_ 1, 14_ 1, 2_ 0, 3_ 1, 16_ 4 \}\) | \(16_0\) | \(\{ 5_ 0, 13_ 0, 16_ 3, 9_ 3, 8_ 1 \}\) | \(6_0\) |
\(\{ 6_ 1, 16_ 0, 10_ 1, 0_ 3, \infty _3\}\) | \(10_0\) | \(\{ 8_ 2, 12_ 1, 15_ 4, 16_ 2, 4_ 2 \}\) | \(15_0\) | \(\{ 2_ 2, 4_ 4, 7_ 2, 3_ 2, 13_ 1 \}\) | \(12_0\) | |
\(\{12_ 2, 1_ 3, 7_ 1, 14_ 0, \infty _4\}\) | \(8_0\) | \(\{ 9_ 4, 0_ 2, 8_ 0, 12_ 4, 7_ 0 \}\) | \(4_0\) | \(\{ 4_ 1, 1_ 0, 9_ 2, 10_ 3, 14_ 2 \}\) | \(11_0\) | |
\(\{ 0_ 0, 2_ 1, 5_ 1, 6_ 2, 12_ 3 \}\) | \(1_0\) | \(\{ 8_ 3, 10_ 2, 14_ 3, 7_ 3, 6_ 4 \}\) | \(0_0\) | \(\{ 7_ 4, 15_ 2, 5_ 4, 1_ 4, 12_ 0 \}\) | \(R\) | |
\(\{ 3_ 2, 7_ 0, 22_ 0, 26_ 2, \infty _0\}\) | \(5_0\) | \(\{ 9_ 3, 3_ 1, 26_ 4, 8_ 0, 23_ 0 \}\) | \(12_0\) | \(\{19_ 0, 12_ 2, 11_ 3, 10_ 4, 27_ 1 \}\) | \(14_0\) | |
\(\{21_ 1, 4_ 3, 14_ 2, 11_ 2, \infty _1\}\) | \(3_0\) | \(\{20_ 3, 26_ 1, 3_ 4, 21_ 0, 6_ 0 \}\) | \(19_0\) | \(\{25_ 0, 28_ 2, 16_ 3, 4_ 4, 5_ 1 \}\) | \(10_0\) | |
\(\{20_ 1, 19_ 3, 23_ 2, 16_ 2, \infty _2\}\) | \(4_0\) | \(\{ 8_ 3, 22_ 1, 7_ 4, 20_ 0, 14_ 0 \}\) | \(13_0\) | \(\{20_ 4, 2_ 4, 5_ 2, 10_ 2, 18_ 0 \}\) | \(28_0\) | |
\(\{ 8_ 1, 25_ 3, 15_ 2, 18_ 2, \infty _3\}\) | \(24_0\) | \(\{ 1_ 0, 11_ 4, 13_ 1, 14_ 1, 26_ 3 \}\) | \(17_0\) | \(\{ 8_ 4, 24_ 4, 2_ 2, 4_ 2, 13_ 0 \}\) | \(9_0\) | |
29 | \(\{ 9_ 1, 10_ 3, 6_ 2, 13_ 2, \infty _4\}\) | \(6_0\) | \(\{12_ 0, 16_ 4, 11_ 1, 23_ 1, 22_ 3 \}\) | \(25_0\) | \(\{ 9_ 4, 27_ 4, 24_ 2, 19_ 2, 11_ 0 \}\) | \(15_0\) |
\(\{12_ 1, 1_ 3, 17_ 4, 20_ 2, 25_ 1 \}\) | \(22_0\) | \(\{28_ 0, 18_ 4, 16_ 1, 15_ 1, 3_ 3 \}\) | \(16_0\) | \(\{21_ 4, 5_ 4, 27_ 2, 25_ 2, 16_ 0 \}\) | \(7_0\) | |
\(\{28_ 1, 12_ 3, 1_ 4, 8_ 2, 10_ 1 \}\) | \(26_0\) | \(\{17_ 0, 13_ 4, 18_ 1, 6_ 1, 7_ 3 \}\) | \(8_0\) | \(\{ 0_ 2, 2_ 3, 23_ 3, 6_ 3, 27_ 3 \}\) | \(1_0\) | |
\(\{17_ 1, 28_ 3, 12_ 4, 9_ 2, 4_ 1 \}\) | \(23_0\) | \(\{10_ 0, 17_ 2, 18_ 3, 19_ 4, 2_ 1 \}\) | \(20_0\) | \(\{ 0_ 3, 6_ 4, 14_ 4, 15_ 4, 23_ 4 \}\) | \(2_0\) | |
\(\{ 1_ 1, 17_ 3, 28_ 4, 21_ 2, 19_ 1 \}\) | \(0_0\) | \(\{ 4_ 0, 1_ 2, 13_ 3, 25_ 4, 24_ 1 \}\) | \(27_0\) | \(\{ 0_ 1, 3_ 0, 7_ 2, 22_ 2, 26_ 0 \}\) | \(11_0\) | |
\(\{ 0_ 4, 24_ 3, 5_ 3, 15_ 3, 14_ 3 \}\) | \(21_0\) | \(\{ 0_ 0, 2_ 0, 24_ 0, 27_ 0, 5_ 0 \}\) | \(18_0\) | \(\{21_ 3, 7_ 1, 22_ 4, 9_ 0, 15_ 0 \}\) | \(R\) |
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Dai, P., Wang, J. & Yin, J. Two series of equitable symbol weight codes meeting the Plotkin bound. Des. Codes Cryptogr. 74, 15–29 (2015). https://doi.org/10.1007/s10623-013-9846-z
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DOI: https://doi.org/10.1007/s10623-013-9846-z