Best Known (102−14, 102, s)-Nets in Base 5
(102−14, 102, 279020)-Net over F5 — Constructive and digital
Digital (88, 102, 279020)-net over F5, using
- 51 times duplication [i] based on digital (87, 101, 279020)-net over F5, using
- net defined by OOA [i] based on linear OOA(5101, 279020, F5, 14, 14) (dual of [(279020, 14), 3906179, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(5101, 1953140, F5, 14) (dual of [1953140, 1953039, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5101, 1953144, F5, 14) (dual of [1953144, 1953043, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(5101, 1953144, F5, 14) (dual of [1953144, 1953043, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(5101, 1953140, F5, 14) (dual of [1953140, 1953039, 15]-code), using
- net defined by OOA [i] based on linear OOA(5101, 279020, F5, 14, 14) (dual of [(279020, 14), 3906179, 15]-NRT-code), using
(102−14, 102, 1009940)-Net over F5 — Digital
Digital (88, 102, 1009940)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5102, 1009940, F5, 14) (dual of [1009940, 1009838, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5102, 1953146, F5, 14) (dual of [1953146, 1953044, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5101, 1953145, F5, 14) (dual of [1953145, 1953044, 15]-code), using
- construction X4 applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(13) ⊂ Ce(11) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5101, 1953145, F5, 14) (dual of [1953145, 1953044, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(5102, 1953146, F5, 14) (dual of [1953146, 1953044, 15]-code), using
(102−14, 102, large)-Net in Base 5 — Upper bound on s
There is no (88, 102, large)-net in base 5, because
- 12 times m-reduction [i] would yield (88, 90, large)-net in base 5, but