MinT - Best Known (89−18, 89, s)-Nets in Base 8

Best Known (89−18, 89, s)-Nets in Base 8

(89−18, 89, 3666)-Net over F₈ — Constructive and digital

Digital (71, 89, 3666)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (58, 76, 3641)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁷⁶, 3641, F₈, 18, 18) (dual of [(3641, 18), 65462, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(8⁷⁶, 32769, F₈, 18) (dual of [32769, 32693, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁷⁶, 32773, F₈, 18) (dual of [32773, 32697, 19]-code), using
        constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
        linear OA(8⁷⁶, 32768, F₈, 18) (dual of [32768, 32692, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(8⁷¹, 32768, F₈, 17) (dual of [32768, 32697, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 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
        OA with only one run / dual of code without redundancy [i]

(89−18, 89, 7282)-Net in Base 8 — Constructive

(71, 89, 7282)-net in base 8, using

8¹ times duplication [i] based on (70, 88, 7282)-net in base 8, using
- base change [i] based on digital (48, 66, 7282)-net over F₁₆, using
  - 16¹ times duplication [i] based on digital (47, 65, 7282)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶⁵, 7282, F₁₆, 18, 18) (dual of [(7282, 18), 131011, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(16⁶⁵, 65538, F₁₆, 18) (dual of [65538, 65473, 19]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁶⁵, 65540, F₁₆, 18) (dual of [65540, 65475, 19]-code), using
        constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
        linear OA(16⁶⁵, 65536, F₁₆, 18) (dual of [65536, 65471, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(16⁶¹, 65536, F₁₆, 17) (dual of [65536, 65475, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(16⁰, 4, F₁₆, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(89−18, 89, 54809)-Net over F₈ — Digital

Digital (71, 89, 54809)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−18, 89, large)-Net in Base 8 — Upper bound on s

There is no (71, 89, large)-net in base 8, because

16 times m-reduction [i] would yield (71, 73, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (89−18, 89, s)-Nets in Base 8

(89−18, 89, 3666)-Net over F8 — Constructive and digital

(89−18, 89, 7282)-Net in Base 8 — Constructive

(89−18, 89, 54809)-Net over F8 — Digital

(89−18, 89, large)-Net in Base 8 — Upper bound on s

(89−18, 89, 3666)-Net over F₈ — Constructive and digital

(89−18, 89, 54809)-Net over F₈ — Digital