Best Known (142−25, 142, s)-Nets in Base 5
(142−25, 142, 6511)-Net over F5 — Constructive and digital
Digital (117, 142, 6511)-net over F5, using
- net defined by OOA [i] based on linear OOA(5142, 6511, F5, 25, 25) (dual of [(6511, 25), 162633, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5142, 78133, F5, 25) (dual of [78133, 77991, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5142, 78141, F5, 25) (dual of [78141, 77999, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(5141, 78126, F5, 25) (dual of [78126, 77985, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(5127, 78126, F5, 23) (dual of [78126, 77999, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5142, 78141, F5, 25) (dual of [78141, 77999, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5142, 78133, F5, 25) (dual of [78133, 77991, 26]-code), using
(142−25, 142, 45418)-Net over F5 — Digital
Digital (117, 142, 45418)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5142, 45418, F5, 25) (dual of [45418, 45276, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5142, 78141, F5, 25) (dual of [78141, 77999, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(5141, 78126, F5, 25) (dual of [78126, 77985, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(5127, 78126, F5, 23) (dual of [78126, 77999, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5142, 78141, F5, 25) (dual of [78141, 77999, 26]-code), using
(142−25, 142, large)-Net in Base 5 — Upper bound on s
There is no (117, 142, large)-net in base 5, because
- 23 times m-reduction [i] would yield (117, 119, large)-net in base 5, but