Best Known (241−26, 241, s)-Nets in Base 3
(241−26, 241, 367921)-Net over F3 — Constructive and digital
Digital (215, 241, 367921)-net over F3, using
- 32 times duplication [i] based on digital (213, 239, 367921)-net over F3, using
- net defined by OOA [i] based on linear OOA(3239, 367921, F3, 26, 26) (dual of [(367921, 26), 9565707, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3239, 4782973, F3, 26) (dual of [4782973, 4782734, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 4782983, F3, 26) (dual of [4782983, 4782744, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3239, 4782969, F3, 26) (dual of [4782969, 4782730, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3225, 4782969, F3, 25) (dual of [4782969, 4782744, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3239, 4782983, F3, 26) (dual of [4782983, 4782744, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3239, 4782973, F3, 26) (dual of [4782973, 4782734, 27]-code), using
- net defined by OOA [i] based on linear OOA(3239, 367921, F3, 26, 26) (dual of [(367921, 26), 9565707, 27]-NRT-code), using
(241−26, 241, 1052225)-Net over F3 — Digital
Digital (215, 241, 1052225)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3241, 1052225, F3, 4, 26) (dual of [(1052225, 4), 4208659, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3241, 1195746, F3, 4, 26) (dual of [(1195746, 4), 4782743, 27]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3240, 1195746, F3, 4, 26) (dual of [(1195746, 4), 4782744, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3240, 4782984, F3, 26) (dual of [4782984, 4782744, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3239, 4782983, F3, 26) (dual of [4782983, 4782744, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3239, 4782969, F3, 26) (dual of [4782969, 4782730, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3225, 4782969, F3, 25) (dual of [4782969, 4782744, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3239, 4782983, F3, 26) (dual of [4782983, 4782744, 27]-code), using
- OOA 4-folding [i] based on linear OA(3240, 4782984, F3, 26) (dual of [4782984, 4782744, 27]-code), using
- 31 times duplication [i] based on linear OOA(3240, 1195746, F3, 4, 26) (dual of [(1195746, 4), 4782744, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3241, 1195746, F3, 4, 26) (dual of [(1195746, 4), 4782743, 27]-NRT-code), using
(241−26, 241, large)-Net in Base 3 — Upper bound on s
There is no (215, 241, large)-net in base 3, because
- 24 times m-reduction [i] would yield (215, 217, large)-net in base 3, but