Best Known (109−15, 109, s)-Nets in Base 5
(109−15, 109, 279019)-Net over F5 — Constructive and digital
Digital (94, 109, 279019)-net over F5, using
- net defined by OOA [i] based on linear OOA(5109, 279019, F5, 15, 15) (dual of [(279019, 15), 4185176, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5109, 1953134, F5, 15) (dual of [1953134, 1953025, 16]-code), using
- 1 times truncation [i] based on linear OA(5110, 1953135, F5, 16) (dual of [1953135, 1953025, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(5110, 1953135, F5, 16) (dual of [1953135, 1953025, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5109, 1953134, F5, 15) (dual of [1953134, 1953025, 16]-code), using
(109−15, 109, 976567)-Net over F5 — Digital
Digital (94, 109, 976567)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5109, 976567, F5, 2, 15) (dual of [(976567, 2), 1953025, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5109, 1953134, F5, 15) (dual of [1953134, 1953025, 16]-code), using
- 1 times truncation [i] based on linear OA(5110, 1953135, F5, 16) (dual of [1953135, 1953025, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(5110, 1953135, F5, 16) (dual of [1953135, 1953025, 17]-code), using
- OOA 2-folding [i] based on linear OA(5109, 1953134, F5, 15) (dual of [1953134, 1953025, 16]-code), using
(109−15, 109, large)-Net in Base 5 — Upper bound on s
There is no (94, 109, large)-net in base 5, because
- 13 times m-reduction [i] would yield (94, 96, large)-net in base 5, but