Best Known (175, 175+23, s)-Nets in Base 3
(175, 175+23, 144939)-Net over F3 — Constructive and digital
Digital (175, 198, 144939)-net over F3, using
- 32 times duplication [i] based on digital (173, 196, 144939)-net over F3, using
- net defined by OOA [i] based on linear OOA(3196, 144939, F3, 23, 23) (dual of [(144939, 23), 3333401, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3196, 1594330, F3, 23) (dual of [1594330, 1594134, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 1594336, F3, 23) (dual of [1594336, 1594140, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3196, 1594336, F3, 23) (dual of [1594336, 1594140, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3196, 1594330, F3, 23) (dual of [1594330, 1594134, 24]-code), using
- net defined by OOA [i] based on linear OOA(3196, 144939, F3, 23, 23) (dual of [(144939, 23), 3333401, 24]-NRT-code), using
(175, 175+23, 398584)-Net over F3 — Digital
Digital (175, 198, 398584)-net over F3, using
- 32 times duplication [i] based on digital (173, 196, 398584)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3196, 398584, F3, 4, 23) (dual of [(398584, 4), 1594140, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3196, 1594336, F3, 23) (dual of [1594336, 1594140, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 4-folding [i] based on linear OA(3196, 1594336, F3, 23) (dual of [1594336, 1594140, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3196, 398584, F3, 4, 23) (dual of [(398584, 4), 1594140, 24]-NRT-code), using
(175, 175+23, large)-Net in Base 3 — Upper bound on s
There is no (175, 198, large)-net in base 3, because
- 21 times m-reduction [i] would yield (175, 177, large)-net in base 3, but