Best Known (248−31, 248, s)-Nets in Base 3
(248−31, 248, 35431)-Net over F3 — Constructive and digital
Digital (217, 248, 35431)-net over F3, using
- 33 times duplication [i] based on digital (214, 245, 35431)-net over F3, using
- net defined by OOA [i] based on linear OOA(3245, 35431, F3, 31, 31) (dual of [(35431, 31), 1098116, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3245, 531466, F3, 31) (dual of [531466, 531221, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 531469, F3, 31) (dual of [531469, 531224, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 531469, F3, 31) (dual of [531469, 531224, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3245, 531466, F3, 31) (dual of [531466, 531221, 32]-code), using
- net defined by OOA [i] based on linear OOA(3245, 35431, F3, 31, 31) (dual of [(35431, 31), 1098116, 32]-NRT-code), using
(248−31, 248, 132868)-Net over F3 — Digital
Digital (217, 248, 132868)-net over F3, using
- 31 times duplication [i] based on digital (216, 247, 132868)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3247, 132868, F3, 4, 31) (dual of [(132868, 4), 531225, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3247, 531472, F3, 31) (dual of [531472, 531225, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3241, 531442, F3, 31) (dual of [531442, 531201, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3217, 531442, F3, 27) (dual of [531442, 531225, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-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([0,15]) ⊂ C([0,13]) [i] based on
- OOA 4-folding [i] based on linear OA(3247, 531472, F3, 31) (dual of [531472, 531225, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3247, 132868, F3, 4, 31) (dual of [(132868, 4), 531225, 32]-NRT-code), using
(248−31, 248, large)-Net in Base 3 — Upper bound on s
There is no (217, 248, large)-net in base 3, because
- 29 times m-reduction [i] would yield (217, 219, large)-net in base 3, but