Best Known (106, 106+18, s)-Nets in Base 3
(106, 106+18, 6563)-Net over F3 — Constructive and digital
Digital (106, 124, 6563)-net over F3, using
- 1 times m-reduction [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
(106, 106+18, 24382)-Net over F3 — Digital
Digital (106, 124, 24382)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3124, 24382, F3, 2, 18) (dual of [(24382, 2), 48640, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3124, 29536, F3, 2, 18) (dual of [(29536, 2), 58948, 19]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3123, 29536, F3, 2, 18) (dual of [(29536, 2), 58949, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3123, 59072, F3, 18) (dual of [59072, 58949, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3122, 59071, F3, 18) (dual of [59071, 58949, 19]-code), using
- construction X4 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(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3122, 59071, F3, 18) (dual of [59071, 58949, 19]-code), using
- OOA 2-folding [i] based on linear OA(3123, 59072, F3, 18) (dual of [59072, 58949, 19]-code), using
- 31 times duplication [i] based on linear OOA(3123, 29536, F3, 2, 18) (dual of [(29536, 2), 58949, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3124, 29536, F3, 2, 18) (dual of [(29536, 2), 58948, 19]-NRT-code), using
(106, 106+18, 7769473)-Net in Base 3 — Upper bound on s
There is no (106, 124, 7769474)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 145557 894028 658285 482628 168475 049901 786667 051718 830925 292853 > 3124 [i]