Best Known (99, 99+16, s)-Nets in Base 5
(99, 99+16, 244144)-Net over F5 — Constructive and digital
Digital (99, 115, 244144)-net over F5, using
- 51 times duplication [i] based on digital (98, 114, 244144)-net over F5, using
- net defined by OOA [i] based on linear OOA(5114, 244144, F5, 16, 16) (dual of [(244144, 16), 3906190, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(5114, 1953152, F5, 16) (dual of [1953152, 1953038, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(5114, 1953157, F5, 16) (dual of [1953157, 1953043, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(5114, 1953157, F5, 16) (dual of [1953157, 1953043, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(5114, 1953152, F5, 16) (dual of [1953152, 1953038, 17]-code), using
- net defined by OOA [i] based on linear OOA(5114, 244144, F5, 16, 16) (dual of [(244144, 16), 3906190, 17]-NRT-code), using
(99, 99+16, 976579)-Net over F5 — Digital
Digital (99, 115, 976579)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5115, 976579, F5, 2, 16) (dual of [(976579, 2), 1953043, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5115, 1953158, F5, 16) (dual of [1953158, 1953043, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5114, 1953157, F5, 16) (dual of [1953157, 1953043, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5114, 1953157, F5, 16) (dual of [1953157, 1953043, 17]-code), using
- OOA 2-folding [i] based on linear OA(5115, 1953158, F5, 16) (dual of [1953158, 1953043, 17]-code), using
(99, 99+16, large)-Net in Base 5 — Upper bound on s
There is no (99, 115, large)-net in base 5, because
- 14 times m-reduction [i] would yield (99, 101, large)-net in base 5, but