Best Known (244−33, 244, s)-Nets in Base 3
(244−33, 244, 11073)-Net over F3 — Constructive and digital
Digital (211, 244, 11073)-net over F3, using
- net defined by OOA [i] based on linear OOA(3244, 11073, F3, 33, 33) (dual of [(11073, 33), 365165, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3244, 177169, F3, 33) (dual of [177169, 176925, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 177171, F3, 33) (dual of [177171, 176927, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3243, 177148, F3, 33) (dual of [177148, 176905, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 23, F3, 1) (dual of [23, 22, 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 X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 177171, F3, 33) (dual of [177171, 176927, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3244, 177169, F3, 33) (dual of [177169, 176925, 34]-code), using
(244−33, 244, 53816)-Net over F3 — Digital
Digital (211, 244, 53816)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3244, 53816, F3, 3, 33) (dual of [(53816, 3), 161204, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3244, 59057, F3, 3, 33) (dual of [(59057, 3), 176927, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3244, 177171, F3, 33) (dual of [177171, 176927, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3243, 177148, F3, 33) (dual of [177148, 176905, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 23, F3, 1) (dual of [23, 22, 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 X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- OOA 3-folding [i] based on linear OA(3244, 177171, F3, 33) (dual of [177171, 176927, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3244, 59057, F3, 3, 33) (dual of [(59057, 3), 176927, 34]-NRT-code), using
(244−33, 244, large)-Net in Base 3 — Upper bound on s
There is no (211, 244, large)-net in base 3, because
- 31 times m-reduction [i] would yield (211, 213, large)-net in base 3, but