Best Known (113−19, 113, s)-Nets in Base 5
(113−19, 113, 8684)-Net over F5 — Constructive and digital
Digital (94, 113, 8684)-net over F5, using
- net defined by OOA [i] based on linear OOA(5113, 8684, F5, 19, 19) (dual of [(8684, 19), 164883, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5113, 78157, F5, 19) (dual of [78157, 78044, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 78160, F5, 19) (dual of [78160, 78047, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(57, 35, F5, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5113, 78160, F5, 19) (dual of [78160, 78047, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5113, 78157, F5, 19) (dual of [78157, 78044, 20]-code), using
(113−19, 113, 72243)-Net over F5 — Digital
Digital (94, 113, 72243)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5113, 72243, F5, 19) (dual of [72243, 72130, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 78156, F5, 19) (dual of [78156, 78043, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(53, 27, F5, 2) (dual of [27, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- linear OA(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5113, 78156, F5, 19) (dual of [78156, 78043, 20]-code), using
(113−19, 113, large)-Net in Base 5 — Upper bound on s
There is no (94, 113, large)-net in base 5, because
- 17 times m-reduction [i] would yield (94, 96, large)-net in base 5, but