Best Known (53, 53+11, s)-Nets in Base 8
(53, 53+11, 419431)-Net over F8 — Constructive and digital
Digital (53, 64, 419431)-net over F8, using
- net defined by OOA [i] based on linear OOA(864, 419431, F8, 11, 11) (dual of [(419431, 11), 4613677, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(864, 2097156, F8, 11) (dual of [2097156, 2097092, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 2097159, F8, 11) (dual of [2097159, 2097095, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(864, 2097159, F8, 11) (dual of [2097159, 2097095, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(864, 2097156, F8, 11) (dual of [2097156, 2097092, 12]-code), using
(53, 53+11, 1242458)-Net over F8 — Digital
Digital (53, 64, 1242458)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(864, 1242458, F8, 11) (dual of [1242458, 1242394, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using
(53, 53+11, large)-Net in Base 8 — Upper bound on s
There is no (53, 64, large)-net in base 8, because
- 9 times m-reduction [i] would yield (53, 55, large)-net in base 8, but