Best Known (213−31, 213, s)-Nets in Base 3
(213−31, 213, 3940)-Net over F3 — Constructive and digital
Digital (182, 213, 3940)-net over F3, using
- 31 times duplication [i] based on digital (181, 212, 3940)-net over F3, using
- net defined by OOA [i] based on linear OOA(3212, 3940, F3, 31, 31) (dual of [(3940, 31), 121928, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3212, 59101, F3, 31) (dual of [59101, 58889, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(3212, 59101, F3, 31) (dual of [59101, 58889, 32]-code), using
- net defined by OOA [i] based on linear OOA(3212, 3940, F3, 31, 31) (dual of [(3940, 31), 121928, 32]-NRT-code), using
(213−31, 213, 22228)-Net over F3 — Digital
Digital (182, 213, 22228)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3213, 22228, F3, 2, 31) (dual of [(22228, 2), 44243, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3213, 29551, F3, 2, 31) (dual of [(29551, 2), 58889, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3213, 59102, F3, 31) (dual of [59102, 58889, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(312, 52, F3, 5) (dual of [52, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- OOA 2-folding [i] based on linear OA(3213, 59102, F3, 31) (dual of [59102, 58889, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3213, 29551, F3, 2, 31) (dual of [(29551, 2), 58889, 32]-NRT-code), using
(213−31, 213, large)-Net in Base 3 — Upper bound on s
There is no (182, 213, large)-net in base 3, because
- 29 times m-reduction [i] would yield (182, 184, large)-net in base 3, but