Best Known (138−20, 138, s)-Nets in Base 3
(138−20, 138, 5908)-Net over F3 — Constructive and digital
Digital (118, 138, 5908)-net over F3, using
- 31 times duplication [i] based on digital (117, 137, 5908)-net over F3, using
- net defined by OOA [i] based on linear OOA(3137, 5908, F3, 20, 20) (dual of [(5908, 20), 118023, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3137, 59080, F3, 20) (dual of [59080, 58943, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, 59085, F3, 20) (dual of [59085, 58948, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-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 Ce(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3137, 59085, F3, 20) (dual of [59085, 58948, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3137, 59080, F3, 20) (dual of [59080, 58943, 21]-code), using
- net defined by OOA [i] based on linear OOA(3137, 5908, F3, 20, 20) (dual of [(5908, 20), 118023, 21]-NRT-code), using
(138−20, 138, 23528)-Net over F3 — Digital
Digital (118, 138, 23528)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3138, 23528, F3, 2, 20) (dual of [(23528, 2), 46918, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 29543, F3, 2, 20) (dual of [(29543, 2), 58948, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3138, 59086, F3, 20) (dual of [59086, 58948, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3137, 59085, F3, 20) (dual of [59085, 58948, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-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 Ce(19) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3137, 59085, F3, 20) (dual of [59085, 58948, 21]-code), using
- OOA 2-folding [i] based on linear OA(3138, 59086, F3, 20) (dual of [59086, 58948, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 29543, F3, 2, 20) (dual of [(29543, 2), 58948, 21]-NRT-code), using
(138−20, 138, large)-Net in Base 3 — Upper bound on s
There is no (118, 138, large)-net in base 3, because
- 18 times m-reduction [i] would yield (118, 120, large)-net in base 3, but