Best Known (213, 213+36, s)-Nets in Base 3
(213, 213+36, 3282)-Net over F3 — Constructive and digital
Digital (213, 249, 3282)-net over F3, using
- 31 times duplication [i] based on digital (212, 248, 3282)-net over F3, using
- t-expansion [i] based on digital (211, 248, 3282)-net over F3, using
- net defined by OOA [i] based on linear OOA(3248, 3282, F3, 37, 37) (dual of [(3282, 37), 121186, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(3248, 59077, F3, 37) (dual of [59077, 58829, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3247, 59076, F3, 37) (dual of [59076, 58829, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3241, 59050, F3, 37) (dual of [59050, 58809, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3221, 59050, F3, 33) (dual of [59050, 58829, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3247, 59076, F3, 37) (dual of [59076, 58829, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(3248, 59077, F3, 37) (dual of [59077, 58829, 38]-code), using
- net defined by OOA [i] based on linear OOA(3248, 3282, F3, 37, 37) (dual of [(3282, 37), 121186, 38]-NRT-code), using
- t-expansion [i] based on digital (211, 248, 3282)-net over F3, using
(213, 213+36, 24487)-Net over F3 — Digital
Digital (213, 249, 24487)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3249, 24487, F3, 2, 36) (dual of [(24487, 2), 48725, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3249, 29545, F3, 2, 36) (dual of [(29545, 2), 58841, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3249, 59090, F3, 36) (dual of [59090, 58841, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 59091, F3, 36) (dual of [59091, 58842, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(3241, 59050, F3, 37) (dual of [59050, 58809, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 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(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 59091, F3, 36) (dual of [59091, 58842, 37]-code), using
- OOA 2-folding [i] based on linear OA(3249, 59090, F3, 36) (dual of [59090, 58841, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(3249, 29545, F3, 2, 36) (dual of [(29545, 2), 58841, 37]-NRT-code), using
(213, 213+36, large)-Net in Base 3 — Upper bound on s
There is no (213, 249, large)-net in base 3, because
- 34 times m-reduction [i] would yield (213, 215, large)-net in base 3, but