Best Known (210, 210+24, s)-Nets in Base 3
(210, 210+24, 398586)-Net over F3 — Constructive and digital
Digital (210, 234, 398586)-net over F3, using
- net defined by OOA [i] based on linear OOA(3234, 398586, F3, 24, 24) (dual of [(398586, 24), 9565830, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3234, 4783032, F3, 24) (dual of [4783032, 4782798, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 4783035, F3, 24) (dual of [4783035, 4782801, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3225, 4782970, F3, 25) (dual of [4782970, 4782745, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(39, 65, F3, 4) (dual of [65, 56, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3234, 4783035, F3, 24) (dual of [4783035, 4782801, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3234, 4783032, F3, 24) (dual of [4783032, 4782798, 25]-code), using
(210, 210+24, 1345924)-Net over F3 — Digital
Digital (210, 234, 1345924)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3234, 1345924, F3, 3, 24) (dual of [(1345924, 3), 4037538, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3234, 1594345, F3, 3, 24) (dual of [(1594345, 3), 4782801, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3234, 4783035, F3, 24) (dual of [4783035, 4782801, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3225, 4782970, F3, 25) (dual of [4782970, 4782745, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(39, 65, F3, 4) (dual of [65, 56, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 3-folding [i] based on linear OA(3234, 4783035, F3, 24) (dual of [4783035, 4782801, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3234, 1594345, F3, 3, 24) (dual of [(1594345, 3), 4782801, 25]-NRT-code), using
(210, 210+24, large)-Net in Base 3 — Upper bound on s
There is no (210, 234, large)-net in base 3, because
- 22 times m-reduction [i] would yield (210, 212, large)-net in base 3, but