I have been reading Delfosse's "Almost-linear time decoding algorithm for topological codes" and I understand how to perform syndrome validation and the posterior erasure decoding in toric codes. Nevertheless, I do not understand how this can work in non-periodic surface codes such as rotated planar codes. Since the Union-Find erasure decoder consists in using the information of the non-trivial syndrome elements in order to find the error-chains, if an error chain finishes on certain boundaries of the rotated planar code, one of the two error-chain boundaries can be missing.
An idea which comes to my mind is using the same tactic adopted by the Minimum Weight Perfect Matching decoder, which consists in considering additional checks adjacent to the boundary of the rotated planar code. These checks would be considered as additional non-trivial syndrome elements, thus affecting the parity of the clusters at which they are involved. Nevertheless, this consideration prevents the use of the erasure decoder.
Consider the plots of the slide:
(1) A rotated planar code experiences an error and, upon measurement, the code outcomes a syndrome.
(2-4) A cluster is produced and grows until it reaches even parity (recall that non-trivial syndrome elements have odd parity and so do the additional boundary checks).
Performing the erasure decoder on the final cluster (4) will not return the maximum likelihood error. I have been looking for a possible answer to this question but have not found anything. Is there an answer? Where could I read about it?
Thank you for your help in advance.