THE NEW CONSTRUCTION OF RANK CODES
T
his research paper, as its title stated “The new construction of
rank codes”, presented a new construction of rank codes, which
defines new codes and includes known codes.
The paper is organized as follows. First, the authors describe known
rank codes. Then, show the new construction and prove that this is the
MRD code. They also give the proof of difference of rank, encoding
and fast encoding algorithms are well-described. In short, the main
purpose of this paper is to demonstrate the way how to produce new
rank codes with fast decoding algorithm.
A set {𝑥1, 𝑥2, … , 𝑥𝑛 } of vectors from 𝑋 𝑛 is called a code with code
distance 𝑑 = min 𝑑(𝑥𝑖 , 𝑥𝑗 ). If the set also forms a 𝑘-dimensional
subspace of 𝑋 𝑛 , then it’s called a linear (𝑛, 𝑘)-code with distance 𝑑.
Such a linear rank metric code always satisfies the Singleton bound
𝑑 ≤ 𝑛 − 𝑘 + 1 . A rank code is also known as an algebraic linear code
over the finite field 𝐺𝐹(𝑞 𝑁 ) similar to Reed-Solomon code.
The only known rank codes are maximum rank distance (MRD)
codes. First, the authors briefly describe the known construction: “Let
ℎ𝑖 ∈ 𝐺𝐹 (𝑞 𝑁 ), 𝑖 = 1, … , 𝑛, be linearly independent over 𝐺𝐹(𝑞). Let
𝑑 ≤ 𝑛. Then the parity-check matrix:
defines MRD code C with the rank distance 𝑑.
The generator matrix G of MRD code C with distance 𝑑 ≤ 𝑛 is the
matrix:
Moving to chap. IV, the authors give the matrix:
with elements ℎ𝑖 ∈ 𝐺𝐹 (𝑞 𝑁 ), 𝑛 ≤ 𝑁, linearly independent over 𝐺𝐹 (𝑞 ).
They’re gonna prove that 𝑯𝑚 is the parity-check matrix for MRD
code provided that 𝑚 and 𝑁 are coprime.
There are a total of four lemmas and four theorems with proofs stated
in the research paper. These fulfill the process of the fast encoding
step. Recall that the fast encoding algorithm for known rank codes
over the subfield 𝐺𝐹 (𝑞 𝑁 ) operates over 𝐺𝐹 (𝑞 𝑁𝑚 ). The authors
change the decoding algorithm to operate over 𝐺𝐹 (𝑞 𝑁 ).
Let g = (𝑔1, … , 𝑔𝑛) be a codeword, e = (𝑒1, … , 𝑒𝑛) be an error
vector, and 𝑦 = 𝑔 + 𝑒 be a received vector.
Let rank norm of the error vector be 𝑙. Then:
Where (𝑒1, … , 𝑒𝑙) ∈ 𝐺𝐹 (𝑞 𝑁 ) are linearly independent over 𝐺𝐹 (𝑞 )
and a matrix Y = (𝑌𝑖𝑗 ) is (𝑙 𝑥 𝑛)-matrix of rank 𝑙 with elements over
𝐺𝐹 (𝑞 ). Then compute the syndrome S:
Elements
are linearly independent over 𝐺𝐹 (𝑞 ).
Therefore, the equation for error detection may be written as the
system of equations:
The goal of decoder is to solve the system of equations for the
minimal 𝑙. This is done via linearized polynomials. The main issue in
the modification of the decoding algorithm is using theorem 3 and 4
given and proved in the research paper.
So that’s every basic things, or fundamentals of building new rank
codes. The rank codes include previously known rank codes as a
partial case, and the fast decoding algorithm is given.