Best Known (113−15, 113, s)-Nets in Base 4
(113−15, 113, 149799)-Net over F4 — Constructive and digital
Digital (98, 113, 149799)-net over F4, using
- 41 times duplication [i] based on digital (97, 112, 149799)-net over F4, using
- net defined by OOA [i] based on linear OOA(4112, 149799, F4, 15, 15) (dual of [(149799, 15), 2246873, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4112, 1048594, F4, 15) (dual of [1048594, 1048482, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4112, 1048597, F4, 15) (dual of [1048597, 1048485, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4112, 1048597, F4, 15) (dual of [1048597, 1048485, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4112, 1048594, F4, 15) (dual of [1048594, 1048482, 16]-code), using
- net defined by OOA [i] based on linear OOA(4112, 149799, F4, 15, 15) (dual of [(149799, 15), 2246873, 16]-NRT-code), using
(113−15, 113, 524299)-Net over F4 — Digital
Digital (98, 113, 524299)-net over F4, using
- 41 times duplication [i] based on digital (97, 112, 524299)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4112, 524299, F4, 2, 15) (dual of [(524299, 2), 1048486, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4112, 1048598, F4, 15) (dual of [1048598, 1048486, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4112, 1048598, F4, 15) (dual of [1048598, 1048486, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4112, 524299, F4, 2, 15) (dual of [(524299, 2), 1048486, 16]-NRT-code), using
(113−15, 113, large)-Net in Base 4 — Upper bound on s
There is no (98, 113, large)-net in base 4, because
- 13 times m-reduction [i] would yield (98, 100, large)-net in base 4, but