Best Known (112, 112+18, s)-Nets in Base 5
(112, 112+18, 217016)-Net over F5 — Constructive and digital
Digital (112, 130, 217016)-net over F5, using
- 52 times duplication [i] based on digital (110, 128, 217016)-net over F5, using
- net defined by OOA [i] based on linear OOA(5128, 217016, F5, 18, 18) (dual of [(217016, 18), 3906160, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5128, 1953144, F5, 18) (dual of [1953144, 1953016, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(15) [i] based on
- OA 9-folding and stacking [i] based on linear OA(5128, 1953144, F5, 18) (dual of [1953144, 1953016, 19]-code), using
- net defined by OOA [i] based on linear OOA(5128, 217016, F5, 18, 18) (dual of [(217016, 18), 3906160, 19]-NRT-code), using
(112, 112+18, 976573)-Net over F5 — Digital
Digital (112, 130, 976573)-net over F5, using
- 51 times duplication [i] based on digital (111, 129, 976573)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5129, 976573, F5, 2, 18) (dual of [(976573, 2), 1953017, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5129, 1953146, F5, 18) (dual of [1953146, 1953017, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5128, 1953145, F5, 18) (dual of [1953145, 1953017, 19]-code), using
- construction X4 applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5128, 1953145, F5, 18) (dual of [1953145, 1953017, 19]-code), using
- OOA 2-folding [i] based on linear OA(5129, 1953146, F5, 18) (dual of [1953146, 1953017, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5129, 976573, F5, 2, 18) (dual of [(976573, 2), 1953017, 19]-NRT-code), using
(112, 112+18, large)-Net in Base 5 — Upper bound on s
There is no (112, 130, large)-net in base 5, because
- 16 times m-reduction [i] would yield (112, 114, large)-net in base 5, but