Best Known (137−19, 137, s)-Nets in Base 5
(137−19, 137, 217016)-Net over F5 — Constructive and digital
Digital (118, 137, 217016)-net over F5, using
- net defined by OOA [i] based on linear OOA(5137, 217016, F5, 19, 19) (dual of [(217016, 19), 4123167, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5137, 1953145, F5, 19) (dual of [1953145, 1953008, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(5137, 1953145, F5, 19) (dual of [1953145, 1953008, 20]-code), using
(137−19, 137, 976572)-Net over F5 — Digital
Digital (118, 137, 976572)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5137, 976572, F5, 2, 19) (dual of [(976572, 2), 1953007, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5137, 1953144, F5, 19) (dual of [1953144, 1953007, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(5137, 1953144, F5, 19) (dual of [1953144, 1953007, 20]-code), using
(137−19, 137, large)-Net in Base 5 — Upper bound on s
There is no (118, 137, large)-net in base 5, because
- 17 times m-reduction [i] would yield (118, 120, large)-net in base 5, but