Best Known (229, 229+20, s)-Nets in Base 3
(229, 229+20, 956598)-Net over F3 — Constructive and digital
Digital (229, 249, 956598)-net over F3, using
- 31 times duplication [i] based on digital (228, 248, 956598)-net over F3, using
- net defined by OOA [i] based on linear OOA(3248, 956598, F3, 22, 20) (dual of [(956598, 22), 21044908, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(3248, 4782991, F3, 2, 20) (dual of [(4782991, 2), 9565734, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3248, 4782994, F3, 2, 20) (dual of [(4782994, 2), 9565740, 21]-NRT-code), using
- trace code [i] based on linear OOA(9124, 2391497, F9, 2, 20) (dual of [(2391497, 2), 4782870, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9124, 4782994, F9, 20) (dual of [4782994, 4782870, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(9120, 4782969, F9, 20) (dual of [4782969, 4782849, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(94, 25, F9, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,9)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(9124, 4782994, F9, 20) (dual of [4782994, 4782870, 21]-code), using
- trace code [i] based on linear OOA(9124, 2391497, F9, 2, 20) (dual of [(2391497, 2), 4782870, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3248, 4782994, F3, 2, 20) (dual of [(4782994, 2), 9565740, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(3248, 4782991, F3, 2, 20) (dual of [(4782991, 2), 9565734, 21]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3248, 956598, F3, 22, 20) (dual of [(956598, 22), 21044908, 21]-NRT-code), using
(229, 229+20, large)-Net over F3 — Digital
Digital (229, 249, large)-net over F3, using
- 38 times duplication [i] based on digital (221, 241, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F3, 20) (dual of [large, large−241, 21]-code), using
- strength reduction [i] based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- strength reduction [i] based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F3, 20) (dual of [large, large−241, 21]-code), using
(229, 229+20, large)-Net in Base 3 — Upper bound on s
There is no (229, 249, large)-net in base 3, because
- 18 times m-reduction [i] would yield (229, 231, large)-net in base 3, but