Best Known (195, 195+33, s)-Nets in Base 3
(195, 195+33, 3692)-Net over F3 — Constructive and digital
Digital (195, 228, 3692)-net over F3, using
- 34 times duplication [i] based on digital (191, 224, 3692)-net over F3, using
- net defined by OOA [i] based on linear OOA(3224, 3692, F3, 33, 33) (dual of [(3692, 33), 121612, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3224, 59073, F3, 33) (dual of [59073, 58849, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3222, 59071, F3, 33) (dual of [59071, 58849, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- 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(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(31, 21, F3, 1) (dual of [21, 20, 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
- 2 times code embedding in larger space [i] based on linear OA(3222, 59071, F3, 33) (dual of [59071, 58849, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3224, 59073, F3, 33) (dual of [59073, 58849, 34]-code), using
- net defined by OOA [i] based on linear OOA(3224, 3692, F3, 33, 33) (dual of [(3692, 33), 121612, 34]-NRT-code), using
(195, 195+33, 23634)-Net over F3 — Digital
Digital (195, 228, 23634)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3228, 23634, F3, 2, 33) (dual of [(23634, 2), 47040, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3228, 29543, F3, 2, 33) (dual of [(29543, 2), 58858, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3228, 59086, F3, 33) (dual of [59086, 58858, 34]-code), using
- 1 times truncation [i] based on linear OA(3229, 59087, F3, 34) (dual of [59087, 58858, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(3229, 59087, F3, 34) (dual of [59087, 58858, 35]-code), using
- OOA 2-folding [i] based on linear OA(3228, 59086, F3, 33) (dual of [59086, 58858, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3228, 29543, F3, 2, 33) (dual of [(29543, 2), 58858, 34]-NRT-code), using
(195, 195+33, large)-Net in Base 3 — Upper bound on s
There is no (195, 228, large)-net in base 3, because
- 31 times m-reduction [i] would yield (195, 197, large)-net in base 3, but