Best Known (74−13, 74, s)-Nets in Base 5
(74−13, 74, 13023)-Net over F5 — Constructive and digital
Digital (61, 74, 13023)-net over F5, using
- 52 times duplication [i] based on digital (59, 72, 13023)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 13023, F5, 13, 13) (dual of [(13023, 13), 169227, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(572, 78139, F5, 13) (dual of [78139, 78067, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 78141, F5, 13) (dual of [78141, 78069, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(571, 78126, F5, 13) (dual of [78126, 78055, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(572, 78141, F5, 13) (dual of [78141, 78069, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(572, 78139, F5, 13) (dual of [78139, 78067, 14]-code), using
- net defined by OOA [i] based on linear OOA(572, 13023, F5, 13, 13) (dual of [(13023, 13), 169227, 14]-NRT-code), using
(74−13, 74, 53397)-Net over F5 — Digital
Digital (61, 74, 53397)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(574, 53397, F5, 13) (dual of [53397, 53323, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(574, 78144, F5, 13) (dual of [78144, 78070, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(572, 78142, F5, 13) (dual of [78142, 78070, 14]-code), using
- construction X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(571, 78126, F5, 13) (dual of [78126, 78055, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(515, 16, F5, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,5)), using
- dual of repetition code with length 16 [i]
- linear OA(51, 16, F5, 1) (dual of [16, 15, 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 X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(572, 78142, F5, 13) (dual of [78142, 78070, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(574, 78144, F5, 13) (dual of [78144, 78070, 14]-code), using
(74−13, 74, large)-Net in Base 5 — Upper bound on s
There is no (61, 74, large)-net in base 5, because
- 11 times m-reduction [i] would yield (61, 63, large)-net in base 5, but