Best Known (188, 188+24, s)-Nets in Base 3
(188, 188+24, 132862)-Net over F3 — Constructive and digital
Digital (188, 212, 132862)-net over F3, using
- 1 times m-reduction [i] based on digital (188, 213, 132862)-net over F3, using
- net defined by OOA [i] based on linear OOA(3213, 132862, F3, 25, 25) (dual of [(132862, 25), 3321337, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3213, 1594345, F3, 25) (dual of [1594345, 1594132, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 1594353, F3, 25) (dual of [1594353, 1594140, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(34, 30, F3, 2) (dual of [30, 26, 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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3213, 1594353, F3, 25) (dual of [1594353, 1594140, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3213, 1594345, F3, 25) (dual of [1594345, 1594132, 26]-code), using
- net defined by OOA [i] based on linear OOA(3213, 132862, F3, 25, 25) (dual of [(132862, 25), 3321337, 26]-NRT-code), using
(188, 188+24, 401951)-Net over F3 — Digital
Digital (188, 212, 401951)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 401951, F3, 3, 24) (dual of [(401951, 3), 1205641, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 531451, F3, 3, 24) (dual of [(531451, 3), 1594141, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3212, 1594353, F3, 24) (dual of [1594353, 1594141, 25]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3210, 1594351, F3, 24) (dual of [1594351, 1594141, 25]-code), using
- construction X4 applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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(24) ⊂ Ce(21) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3210, 1594351, F3, 24) (dual of [1594351, 1594141, 25]-code), using
- OOA 3-folding [i] based on linear OA(3212, 1594353, F3, 24) (dual of [1594353, 1594141, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 531451, F3, 3, 24) (dual of [(531451, 3), 1594141, 25]-NRT-code), using
(188, 188+24, large)-Net in Base 3 — Upper bound on s
There is no (188, 212, large)-net in base 3, because
- 22 times m-reduction [i] would yield (188, 190, large)-net in base 3, but