Best Known (128, 128+21, s)-Nets in Base 3
(128, 128+21, 5908)-Net over F3 — Constructive and digital
Digital (128, 149, 5908)-net over F3, using
- 32 times duplication [i] based on digital (126, 147, 5908)-net over F3, using
- net defined by OOA [i] based on linear OOA(3147, 5908, F3, 21, 21) (dual of [(5908, 21), 123921, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3147, 59081, F3, 21) (dual of [59081, 58934, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3147, 59085, F3, 21) (dual of [59085, 58938, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3147, 59085, F3, 21) (dual of [59085, 58938, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3147, 59081, F3, 21) (dual of [59081, 58934, 22]-code), using
- net defined by OOA [i] based on linear OOA(3147, 5908, F3, 21, 21) (dual of [(5908, 21), 123921, 22]-NRT-code), using
(128, 128+21, 29545)-Net over F3 — Digital
Digital (128, 149, 29545)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3149, 29545, F3, 2, 21) (dual of [(29545, 2), 58941, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3149, 59090, F3, 21) (dual of [59090, 58941, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(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]
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3149, 59090, F3, 21) (dual of [59090, 58941, 22]-code), using
(128, 128+21, large)-Net in Base 3 — Upper bound on s
There is no (128, 149, large)-net in base 3, because
- 19 times m-reduction [i] would yield (128, 130, large)-net in base 3, but