Finite-Field Size

Having found the set of orthogonal permutations, we need to specify the coefficients in the parities such that the code is MDS.

Consider the(m+3,m+1)codeC constructed by Theorem 4.3 and the vectors{ei}m_i=0. Let

Fbe the finite field in use, where the information in rowi, column jisai,j ∈ F. Let its row and zigzag coefficients beαi,j,βi,j ∈F, respectively. For a row setRu={au,0,au,1, . . . ,au,m}, the row parity isru= ∑mj=0αu,jau,j. For a zigzag setZu= {au,0,au+e1,1, . . . ,au+em,m}, the zigzag parity is zu=∑mj=0βu+ej,jau+ej,j.

Recall that the(m+3,m+1)code is indeed MDS iff we can recover the information from up to2columns erasures. It is clear that none of the coefficientsαi,j,βi,j can be zero. Moreover, if we assign all the coefficients as αi,j = βi,j = 1we get that in an erasure of two systematic columns the set of equations derived from the parity columns are linearly dependent and thus not solvable (the sum of the equations from the row parity and the sum of those from the zigzag parity will both be the sum of the entire information array). Therefore the coefficients need to be from a field with more than one nonzero element, thus a field of size at least3is necessary.

to special structure of the code. We are going to take advantage of it and assign the coefficients accordingly. SurprisinglyF3is sufficient to correct two erasures.

Construction 4.5 For the codeC in Theorem4.3 overF₃, defineuj = ∑j_l=0el for0 ≤ j ≤ m. Assign row coefficients asαi,j =1for alli,j, and zigzag coefficients as

βi,j =2i·uj (4.3)

where i = (i1, . . . ,im) is represented in binary and the calculation of the inner product in the

exponent is done overF₂.

The coefficients in Figure 4.4 are assigned by Construction 4.5. For example,

β3,1 =23·u1 =2(1,1)·(1,0) =21 =2.

β3,2 =23·u2 =2(1,1)·(1,1) =20 =1.

One can check that the code can tolerate any two erasures and hence is MDS. The following theorem shows that the construction is MDS.

Theorem 4.6 Construction4.5is an(m+3,m+1)MDS code with optimal finite-field size of3. Proof: It is easy to see that if at least one of the two erased columns is a parity column then we can recover the information. Hence we only need to show that we can recover from an erasure of any two systematic columnsi,j ∈[0,m],i < j. In this scenario, we access the entire remaining information in the array. For r ∈ [0, 2m−1]set r0 = r+ei+ej, and recall that ar,i ∈ Zl iff

l = r+ei, thusar,i,ar0_,j ∈ Z_r+e

i andar,j,ar0,i ∈ Zr+ej. From the two parity columns we need to

solve the following equations         1 1 0 0 0 0 1 1 βr,i 0 0 βr0_,j 0 β_r,j β_r0_,i 0                 ar,i ar,j ar0_,i ar0_,j         =         y1 y2 y3 y4         . (4.4)

Herey1, . . . ,y4 ∈ F3are the differences of the corresponding parity elements (the r-th, r0-th row

the row/zigzag sets. This set of equations is solvable iff

β_r,iβ_r0_,i 6= β_r,jβ_r0_,j. (4.5)

Note that the multiplicative group ofF3\0is isomorphic to the additive group ofF2, hence mul-

tiplying two elements in F3\0 is equivalent to summing up their exponent in F2 when they are

represented as a power of the primitive element of the fieldF3. For columns0 ≤ i < j≤ mand

rowsr,r0defined above, we have

βr,iβr0_,i =2r·ui+r 0_·u

i ₌₂(r+r0)·ui _{= 2}(ei+ej)·∑il=0el ₌₂ei·ei _=2.

However in the same manner we derive that

β_r,jβ_r0_,j =2(r+r 0_)·u

j ₌₂(ei+ej)·∑jl=0el ₌₂ei·ei+ej·ej ₌₂0_=1.

Hence (4.5) is satisfied and the code is MDS.

Remark: The above proof shows that βr,i 6= βr0_,i, and βr,j = βr0_,j fori < j. And (4.5) is a

necessary and sufficient condition for correcting erasure of columnsiandjsuch thatvi 6= vj. It should be noted that in practice it is more convenient to use finite field GF(2s)for some integers. In fact we can use a field of size4by simply modifying (4.3) to

βi,j =ci·uj,

wherecis a primitive element inGF(4)and computations are done overF2 in the exponent. It is

obvious that this will not affect the proof of Theorem 4.6.

In addition to optimal ratio and optimal field size, we will show in the next section that the code in Theorem 4.3 is also of optimal array size, namely, it has the maximum number of columns, given the number of rows.