Best Known (224, 224+20, s)-Nets in Base 3
(224, 224+20, 956595)-Net over F3 — Constructive and digital
Digital (224, 244, 956595)-net over F3, using
- 34 times duplication [i] based on digital (220, 240, 956595)-net over F3, using
- net defined by OOA [i] based on linear OOA(3240, 956595, F3, 22, 20) (dual of [(956595, 22), 21044850, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(3240, 4782976, F3, 2, 20) (dual of [(4782976, 2), 9565712, 21]-NRT-code), using
- trace code [i] based on linear OOA(9120, 2391488, F9, 2, 20) (dual of [(2391488, 2), 4782856, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9120, 4782976, F9, 20) (dual of [4782976, 4782856, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [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(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(9120, 4782976, F9, 20) (dual of [4782976, 4782856, 21]-code), using
- trace code [i] based on linear OOA(9120, 2391488, F9, 2, 20) (dual of [(2391488, 2), 4782856, 21]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OOA(3240, 4782976, F3, 2, 20) (dual of [(4782976, 2), 9565712, 21]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3240, 956595, F3, 22, 20) (dual of [(956595, 22), 21044850, 21]-NRT-code), using
(224, 224+20, large)-Net over F3 — Digital
Digital (224, 244, large)-net over F3, using
- 33 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
(224, 224+20, large)-Net in Base 3 — Upper bound on s
There is no (224, 244, large)-net in base 3, because
- 18 times m-reduction [i] would yield (224, 226, large)-net in base 3, but