raptorq/RFC6330_ERRATA.md

Errata and optimizations for RFC6330

The following is a list of unofficial errata and potential optimizations, which I discovered while writing this library.

1. Section 5.4.2.2 says "There are at most L steps in the first phase". This should say, "at most L - P steps". This is clear from the fact that the end condition is i + u = L, u is initialized to P, and i is incremented by 1 at each step.
2. Section 5.4.2.2 refers to choosing HDPC rows after non-HDPC rows. However, this can be strengthened to "HDPC rows should never be chosen". This follows from the fact that at the end of the first phase, u rows, will not have been chosen. Therefore if P >= H, HDPC rows will never be chosen, since the first phase will end before they can be chosen. Proof that P >= H:
   P >= H
   L - W >= H (substitute P)
   K' + S + H - W >= H (substitute L)
   K' + S >= W (subtract H and add W to both sides)
   K' + S >= W, can be verified empirically from the systematic constants in Table 2
3. Optimization: HDPC rows do not need to be copied into X. At the beginning of the second phase, X is truncated to be i by i. It follows from errata 2 that none of these can be HDPC rows, since the same row and column swaps are performed on X as on A.
4. Optimization: all non-HDPC rows have only the values zero and one, for the entire coding process. Proof:
   1. at construction the constraint matrix has only zeros and ones in non-HDPC rows
   2. during the first phase, only non-HDPC rows are added to other rows (see errata 2), since these contain only ones or zeros, and are used in elimination they will never be multiplied by a beta > 1, and therefore will never create a value > 1 in a non-HDPC row.
   3. at the beginning of the second phase, U_lower is the only part of the matrix which can have values > 1. This follows from the fact that: a) the submatrix V no longer exists. b) no HDPC rows can exist in the identity submatrix I (see errata 2). The second phase ends either in failure, or with U_lower equal to the identity matrix. During the second phase, no operations are performed on preceding U_lower. Therefore, at the end of the second phase A contains only binary values.
   4. the third phase does not introduce non-binary values, given that none already exist, because matrix multiplication over GF(256) with binary values cannot produce a non-binary value.
   5. the fourth phase does not introduce non-binary values unless one already exists. The variable "b" referenced in the specification can only be one, since the matrix is already binary.
   6. the fifth phase already cannot introduce non-binary values, because the matrix is already binary
5. Optimization: the matrix A is binary after the second phase. (see proof in errata 4)
6. Section 5.4.2.5 says "For each of the first i rows of U_upper, do the following: if the row has a nonzero entry at position j, and if the value of that nonzero entry is b, then add to this row b times row j of I_u". This should say "For each of the first i rows of U_upper, do the following: if the row has a one-valued entry at position j, then add to this row, row j of I_u". This follows from the fact that all nonzero values are one (see errata 5).
7. Section 5.4.2.6 says:

   For j from 1 to i, perform the following operations:
      1.  If A[j,j] is not one, then divide row j of A by A[j,j].
      2.  For l from 1 to j-1, if A[j,l] is nonzero, then add A[j,l]
          multiplied with row l of A to row j of A.
   

   Because A is a binary matrix, this can be simplified to:

   For j from 1 to i, perform the following operations:
      For l from 1 to j-1, if A[j,l] is one, then add row l of A to row j of A.
   

8. Section 5.4.2.2 says "If r = 2 and there is no row with exactly 2 ones in V, then choose any row with exactly 2 nonzeros in V." It follows from Errata 2 & 4 that this can be ignored as it will never happen.
9. The row operations performed in the fifth phase are the same operations performed in the first phase on the i x i submatrix which is converted to the identity. This follows from the fact that this i x i submatrix is equal (including row & column swaps) after the third phase: After this operation, the submatrix of A consisting of the intersection of the first i rows and columns equals to X (note that X is equal to the original A matrix), and the fact that the fourth phase does not modify this submatrix because it only adds zero. Therefore, the elimination performed in the fifth phase is the same as that performed on the i x i submatrix during the first phase.
10. The row operations performed in the third phase are the reverse of the operations performed on the i x i submatrix in the first phase. This can be seen from the fact that this i x i submatrix has been reset to X after the third phase After this operation, the submatrix of A consisting of the intersection of the first i rows and columns equals to X which is accomplished by undoing (reversing) all the row operations performed in the first phase, since the second phase does not modify this section of the matrix.
11. If the row operations from the first phase are recorded, then the first i columns of A may be discarded after the end of the first phase. This follows from errata 9 & 10 which guarantee that the third and fifth phases can be performed using this record, and the fact that the second and fourth phases only operate on the U section of A (the columns after i).
12. To improve performance the connected components of the graph for the "r = 2" selection step of the first phase should be calculated once, and the connected components stored. The connected components can then be updated whenever a new row with 2 ones is created or deleted, during the first phase.