Best Known (110−12, 110, s)-Nets in Base 7
(110−12, 110, 1921870)-Net over F7 — Constructive and digital
Digital (98, 110, 1921870)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (14, 20, 269)-net over F7, using
- net defined by OOA [i] based on linear OOA(720, 269, F7, 6, 6) (dual of [(269, 6), 1594, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(720, 807, F7, 6) (dual of [807, 787, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(720, 808, F7, 6) (dual of [808, 788, 7]-code), using
- construction XX applied to C1 = C([130,134]), C2 = C([129,133]), C3 = C1 + C2 = C([130,133]), and C∩ = C1 ∩ C2 = C([129,134]) [i] based on
- linear OA(716, 800, F7, 5) (dual of [800, 784, 6]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {130,131,132,133,134}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(716, 800, F7, 5) (dual of [800, 784, 6]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {129,130,131,132,133}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(720, 800, F7, 6) (dual of [800, 780, 7]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {129,130,…,134}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(712, 800, F7, 4) (dual of [800, 788, 5]-code), using the BCH-code C(I) with length 800 | 74−1, defining interval I = {130,131,132,133}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([130,134]), C2 = C([129,133]), C3 = C1 + C2 = C([130,133]), and C∩ = C1 ∩ C2 = C([129,134]) [i] based on
- discarding factors / shortening the dual code based on linear OA(720, 808, F7, 6) (dual of [808, 788, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(720, 807, F7, 6) (dual of [807, 787, 7]-code), using
- net defined by OOA [i] based on linear OOA(720, 269, F7, 6, 6) (dual of [(269, 6), 1594, 7]-NRT-code), using
- digital (78, 90, 1921601)-net over F7, using
- net defined by OOA [i] based on linear OOA(790, 1921601, F7, 14, 12) (dual of [(1921601, 14), 26902324, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(790, 5764804, F7, 2, 12) (dual of [(5764804, 2), 11529518, 13]-NRT-code), using
- trace code [i] based on linear OOA(4945, 2882402, F49, 2, 12) (dual of [(2882402, 2), 5764759, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4945, 5764804, F49, 12) (dual of [5764804, 5764759, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4945, 5764805, F49, 12) (dual of [5764805, 5764760, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4941, 5764801, F49, 11) (dual of [5764801, 5764760, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4945, 5764805, F49, 12) (dual of [5764805, 5764760, 13]-code), using
- OOA 2-folding [i] based on linear OA(4945, 5764804, F49, 12) (dual of [5764804, 5764759, 13]-code), using
- trace code [i] based on linear OOA(4945, 2882402, F49, 2, 12) (dual of [(2882402, 2), 5764759, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(790, 5764804, F7, 2, 12) (dual of [(5764804, 2), 11529518, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(790, 1921601, F7, 14, 12) (dual of [(1921601, 14), 26902324, 13]-NRT-code), using
- digital (14, 20, 269)-net over F7, using
(110−12, 110, large)-Net over F7 — Digital
Digital (98, 110, large)-net over F7, using
- 71 times duplication [i] based on digital (97, 109, large)-net over F7, using
- t-expansion [i] based on digital (94, 109, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176804 | 718−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- t-expansion [i] based on digital (94, 109, large)-net over F7, using
(110−12, 110, large)-Net in Base 7 — Upper bound on s
There is no (98, 110, large)-net in base 7, because
- 10 times m-reduction [i] would yield (98, 100, large)-net in base 7, but