Coding Theory by Jürgen Müller

By Jürgen Müller

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Hinweis. Man z¨ ahle {[v, w] ∈ C × Fn2 ; d(v, w) = 2} mittels Double-Counting. 15) Aufgabe: Ausl¨ oschungen. Es sei C ein nicht-trivialer Block-Code. Geht bei der Daten¨ ubertragung ein Eintrag eines Wortes verloren, so wird er als Ausl¨ oschung markiert; dies ist also ein Fehler mit bekannter Position. F¨ ur e, g ∈ N0 zeige man: Der Code C kann genau dann gleichzeitig e Fehler und g Ausl¨oschungen korrigieren, wenn 2e + g + 1 ≤ d(C) gilt. 16) Aufgabe: Parameter f¨ ur Codes. F¨ ur n ∈ N betrachte man einen (n, q n−3 , 3)-Code u ¨ber einem Alphabet mit q Elementen.

1 1 1 . 1     . 1 1 1 . 1 1     . 1 . . 1 1 1     . 1 1 . 1 . .    . 1 1 1 . 1 1 . A12 := 1 . . 1 1 1 .     1 . 1 . . 1 1     1 . 1 1 . 1 .     1 1 . 1 . . 1     1 1 . 1 1 . 1 .    1 1 1 . 1 1 . 1 von Zeilenvektoren gegebenen . 1 1 . 1 . 1             . 1 . 1 1 1 1 . 1 . . 1 1 1 . 1 . 1 1 . ⊆ F72 ,            . 1 . 1 1 . 1 1 . 1 .                     ⊆ F11 2 .                    Außerdem seien 17 + A8 := {17 + v ∈ F72 ; v ∈ A8 } und 111 + A12 := {111 + v ∈ F11 2 ; v ∈ A12 } die jeweiligen Komplemente, sowie B8 := A8 ∪ (17 + A8 ) und B11 := A12 ∪ (111 + A12 ).

Letting P := {0, −16, −15} we get P + P = {0, . . , 14} ∪ {16, . . , 29} ∪ {32, . . , 44} ⊆ O, hence C is associated with P + P as well. Moreover, from P = {0, −1, 7} · 16 ⊆ Z127 and P := {−1, 0, 7} ⊆ {−1, . . , 7} =: P , hence |P | − |P | = 6, we get rkFq (ζn ) (H(P )I ) ≥ |I| whenever |I| ≤ 2, and rkFq (ζn ) (H(P )I ) ≥ 2 whenever 3 ≤ |I| ≤ 7, and rkFq (ζn ) (H(P )I ) ≥ |I| − 6 whenever 8 ≤ |I| ≤ 9, and rkFq (ζn ) (H(P )I ) = 3 whenever |I| ≥ 10. Thus the van-Lint-Wilson bound yields d ≥ 30.

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