Best Known (127−19, 127, s)-Nets in Base 3
(127−19, 127, 6563)-Net over F3 — Constructive and digital
Digital (108, 127, 6563)-net over F3, using
- 32 times duplication [i] based on digital (106, 125, 6563)-net over F3, using
- net defined by OOA [i] based on linear OOA(3125, 6563, F3, 19, 19) (dual of [(6563, 19), 124572, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3125, 59068, F3, 19) (dual of [59068, 58943, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 59073, F3, 19) (dual of [59073, 58948, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3125, 59073, F3, 19) (dual of [59073, 58948, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3125, 59068, F3, 19) (dual of [59068, 58943, 20]-code), using
- net defined by OOA [i] based on linear OOA(3125, 6563, F3, 19, 19) (dual of [(6563, 19), 124572, 20]-NRT-code), using
(127−19, 127, 19692)-Net over F3 — Digital
Digital (108, 127, 19692)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3127, 19692, F3, 3, 19) (dual of [(19692, 3), 58949, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3127, 59076, F3, 19) (dual of [59076, 58949, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(3101, 59050, F3, 15) (dual of [59050, 58949, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 3-folding [i] based on linear OA(3127, 59076, F3, 19) (dual of [59076, 58949, 20]-code), using
(127−19, 127, large)-Net in Base 3 — Upper bound on s
There is no (108, 127, large)-net in base 3, because
- 17 times m-reduction [i] would yield (108, 110, large)-net in base 3, but