Best Known (48−11, 48, s)-Nets in Base 7
(48−11, 48, 3363)-Net over F7 — Constructive and digital
Digital (37, 48, 3363)-net over F7, using
- 71 times duplication [i] based on digital (36, 47, 3363)-net over F7, using
- net defined by OOA [i] based on linear OOA(747, 3363, F7, 11, 11) (dual of [(3363, 11), 36946, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(747, 16816, F7, 11) (dual of [16816, 16769, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(747, 16818, F7, 11) (dual of [16818, 16771, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(747, 16818, F7, 11) (dual of [16818, 16771, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(747, 16816, F7, 11) (dual of [16816, 16769, 12]-code), using
- net defined by OOA [i] based on linear OOA(747, 3363, F7, 11, 11) (dual of [(3363, 11), 36946, 12]-NRT-code), using
(48−11, 48, 16821)-Net over F7 — Digital
Digital (37, 48, 16821)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(748, 16821, F7, 11) (dual of [16821, 16773, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(747, 16819, F7, 11) (dual of [16819, 16772, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(711, 12, F7, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,7)), using
- dual of repetition code with length 12 [i]
- linear OA(71, 12, F7, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(747, 16820, F7, 10) (dual of [16820, 16773, 11]-code), using Gilbert–Varšamov bound and bm = 747 > Vbs−1(k−1) = 2984 755571 977452 274395 337031 019568 277719 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 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(747, 16819, F7, 11) (dual of [16819, 16772, 12]-code), using
- construction X with Varšamov bound [i] based on
(48−11, 48, large)-Net in Base 7 — Upper bound on s
There is no (37, 48, large)-net in base 7, because
- 9 times m-reduction [i] would yield (37, 39, large)-net in base 7, but