Best Known (193, 193+33, s)-Nets in Base 3
(193, 193+33, 3692)-Net over F3 — Constructive and digital
Digital (193, 226, 3692)-net over F3, using
- 32 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
(193, 193+33, 21963)-Net over F3 — Digital
Digital (193, 226, 21963)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3226, 21963, F3, 2, 33) (dual of [(21963, 2), 43700, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3226, 29538, F3, 2, 33) (dual of [(29538, 2), 58850, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3226, 59076, F3, 33) (dual of [59076, 58850, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 59077, F3, 33) (dual of [59077, 58851, 34]-code), using
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ 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(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31, 24, F3, 1) (dual of [24, 23, 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(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 59077, F3, 33) (dual of [59077, 58851, 34]-code), using
- OOA 2-folding [i] based on linear OA(3226, 59076, F3, 33) (dual of [59076, 58850, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3226, 29538, F3, 2, 33) (dual of [(29538, 2), 58850, 34]-NRT-code), using
(193, 193+33, large)-Net in Base 3 — Upper bound on s
There is no (193, 226, large)-net in base 3, because
- 31 times m-reduction [i] would yield (193, 195, large)-net in base 3, but