Best Known (96, 96+15, s)-Nets in Base 5
(96, 96+15, 279020)-Net over F5 — Constructive and digital
Digital (96, 111, 279020)-net over F5, using
- 51 times duplication [i] based on digital (95, 110, 279020)-net over F5, using
- net defined by OOA [i] based on linear OOA(5110, 279020, F5, 15, 15) (dual of [(279020, 15), 4185190, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5110, 1953141, F5, 15) (dual of [1953141, 1953031, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5110, 1953145, F5, 15) (dual of [1953145, 1953035, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(5109, 1953126, F5, 15) (dual of [1953126, 1953017, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(591, 1953126, F5, 13) (dual of [1953126, 1953035, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(51, 19, F5, 1) (dual of [19, 18, 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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5110, 1953145, F5, 15) (dual of [1953145, 1953035, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5110, 1953141, F5, 15) (dual of [1953141, 1953031, 16]-code), using
- net defined by OOA [i] based on linear OOA(5110, 279020, F5, 15, 15) (dual of [(279020, 15), 4185190, 16]-NRT-code), using
(96, 96+15, 1163334)-Net over F5 — Digital
Digital (96, 111, 1163334)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5111, 1163334, F5, 15) (dual of [1163334, 1163223, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5111, 1953147, F5, 15) (dual of [1953147, 1953036, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5110, 1953146, F5, 15) (dual of [1953146, 1953036, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(5109, 1953126, F5, 15) (dual of [1953126, 1953017, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(591, 1953126, F5, 13) (dual of [1953126, 1953035, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
- dual of repetition code with length 20 [i]
- linear OA(51, 20, F5, 1) (dual of [20, 19, 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([0,7]) ⊂ C([0,6]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5110, 1953146, F5, 15) (dual of [1953146, 1953036, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5111, 1953147, F5, 15) (dual of [1953147, 1953036, 16]-code), using
(96, 96+15, large)-Net in Base 5 — Upper bound on s
There is no (96, 111, large)-net in base 5, because
- 13 times m-reduction [i] would yield (96, 98, large)-net in base 5, but