Best Known (89, 106, s)-Nets in Base 5
(89, 106, 48829)-Net over F5 — Constructive and digital
Digital (89, 106, 48829)-net over F5, using
- 51 times duplication [i] based on digital (88, 105, 48829)-net over F5, using
- net defined by OOA [i] based on linear OOA(5105, 48829, F5, 17, 17) (dual of [(48829, 17), 829988, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5105, 390633, F5, 17) (dual of [390633, 390528, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(5105, 390633, F5, 17) (dual of [390633, 390528, 18]-code), using
- net defined by OOA [i] based on linear OOA(5105, 48829, F5, 17, 17) (dual of [(48829, 17), 829988, 18]-NRT-code), using
(89, 106, 195317)-Net over F5 — Digital
Digital (89, 106, 195317)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5106, 195317, F5, 2, 17) (dual of [(195317, 2), 390528, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5106, 390634, F5, 17) (dual of [390634, 390528, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5105, 390633, F5, 17) (dual of [390633, 390528, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5105, 390633, F5, 17) (dual of [390633, 390528, 18]-code), using
- OOA 2-folding [i] based on linear OA(5106, 390634, F5, 17) (dual of [390634, 390528, 18]-code), using
(89, 106, large)-Net in Base 5 — Upper bound on s
There is no (89, 106, large)-net in base 5, because
- 15 times m-reduction [i] would yield (89, 91, large)-net in base 5, but