Best Known (109−17, 109, s)-Nets in Base 7
(109−17, 109, 102957)-Net over F7 — Constructive and digital
Digital (92, 109, 102957)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (83, 100, 102944)-net over F7, using
- net defined by OOA [i] based on linear OOA(7100, 102944, F7, 17, 17) (dual of [(102944, 17), 1749948, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7100, 823553, F7, 17) (dual of [823553, 823453, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(7100, 823559, F7, 17) (dual of [823559, 823459, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(799, 823544, F7, 17) (dual of [823544, 823445, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(785, 823544, F7, 15) (dual of [823544, 823459, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(71, 15, F7, 1) (dual of [15, 14, 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 C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7100, 823559, F7, 17) (dual of [823559, 823459, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7100, 823553, F7, 17) (dual of [823553, 823453, 18]-code), using
- net defined by OOA [i] based on linear OOA(7100, 102944, F7, 17, 17) (dual of [(102944, 17), 1749948, 18]-NRT-code), using
- digital (1, 9, 13)-net over F7, using
(109−17, 109, 823591)-Net over F7 — Digital
Digital (92, 109, 823591)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7109, 823591, F7, 17) (dual of [823591, 823482, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(799, 823543, F7, 17) (dual of [823543, 823444, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(7106, 823588, F7, 16) (dual of [823588, 823482, 17]-code), using Gilbert–Varšamov bound and bm = 7106 > Vbs−1(k−1) = 19559 481431 446353 245831 493276 788957 679380 588391 039563 056234 797349 577467 625004 211603 961431 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 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(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
- construction X with Varšamov bound [i] based on
(109−17, 109, large)-Net in Base 7 — Upper bound on s
There is no (92, 109, large)-net in base 7, because
- 15 times m-reduction [i] would yield (92, 94, large)-net in base 7, but