A set ${\fancyscript{K}}$ in PG(r, 4), r ≥ 2, is odd if every line meets ${\fancyscript{K}}$ in an odd number of points. An odd set ${\fancyscript{K}}$ in PG(r, 4), r ≥ 3, is FH-free if there is no plane meeting ${\fancyscript{K}}$ in a Fano plane or in a non-singular Hermitian curve. We prove that an odd set ${\fancyscript{K}}$ contains a hyperplane of PG(r, 4) if and only if ${\fancyscript{K}}$ is FH-free. As an application to coding theory, a new extension theorem for quaternary linear codes is given.

A characterization of some odd sets in projective spaces of order 4 and the extendability of quaternary linear codes

On the l-extendability of quaternary linear codes

On the extendability of quaternary linear codes with four weights modulo 16

A set $${\fancyscript{K}}$$
 in $$\hbox {PG}(r,4), r \ge 2$$
 , is odd if every line meets $${\fancyscript{K}}$$
 in an odd number of points. We show there are exactly 45 inequivalent odd sets in $$\hbox {PG}(4,4)$$
 up to projective equivalence. As an application to coding theory, a new sufficient condition for the extendability of quaternary linear codes is given.

Classification of the odd sets in $$\hbox {PG}(4,4)$$ PG ( 4 , 4 ) and its application to coding theory

We give four new extension theorems for linear codes over Fq :( a) Forq = 2 h , h ≥ 3, every [n, k, d]q code with d odd whose weights are congruent to 0o rd (mod q/2) is extendable. (b) For q =2 h , h ≥ 3, every [n, k, d]q code with gcd(d, q) = 2 whose weights are congruent to 0 or d (mod q )i s doubly extendable. (c) For integers h, m, t with 0 ≤ m<t ≤ h and prime p ,e very [n, k, d]q code with gcd(d, q )= p m and q = p h is extendable if �

/pdf/some-generalizations-of-extension-theorems-for-linear-codes-4a27pe96xw.pdf

Taichiro Tanaka

Papers

Classification of the odd sets in $$\hbox {PG}(4,4)$$ PG ( 4 , 4 ) and its application to coding theory

A characterization of some odd sets in projective spaces of order 4 and the extendability of quaternary linear codes

On the l-extendability of quaternary linear codes

Some generalizations of extension theorems for linear codes over finite fields