In large-scale distributed storage systems, erasure coding is employed to ensure reliability against disk failures. Recent work by Kadekodi et al. demonstrates that adapting code parameters to varying disk failure rates can lead to significant storage savings without compromising reliability. Such adaptations, known ascode conversions, motivate the design ofconvertible codes, which enable efficient transformations between codes of different parameters. In this work, we study the setting in which λ codewords of an initial [nI=kI+rI,kI] MDS code are merged into a single codeword of a final [nF= λkI+rF,kF= λkI] MDS code. We begin by presenting three constructions that achieve optimalaccess cost, defined as the total number of disks accessed during the conversion process. The first two constructions apply when λ ≤rIand impose specific divisibility conditions onrIand the field sizeq. These schemes minimize both the per-symbol and the overall access cost. The third construction, which builds on a prior scheme by Kong, achieves minimal access cost while supporting arbitrary parameter regimes. All three constructions require field sizes that are linear in the final code length, and notably, the third construction achieves a field size that matches the lower bound implied by the MDS conjecture in almost all cases. In addition, we propose a construction that optimizes thebandwidth cost, defined as the total number of symbols transmitted during conversion. This scheme is a refinement of Maturana and Rashmi’s bandwidth-optimal construction based on the piggybacking framework, and achieves reduced sub-packetization.The code will be available at https://github.com/ZhilongNiu/SDMoMFE.

In large-scale distributed storage systems, erasure coding provides reliability against disk failures. Recent work by Kadekodi et al. demonstrates that adapting code parameters to varying disk failure rates can lead to storage savings, without compromising reliability. During such adaptations (also called code conversions), data encoded with multiple codewords of an initial [$n^{I}, k^{I}$] code are transformed into multiple codewords of a final$\left[n^{F}, k^{F}\right]$code. The convertible codes framework aims to design initial and final codes to enable resource-efficient conversions. In this paper, we study the scenario where$\lambda \geq 2$codewords of an initial$\left[n^{I}=k^{I}+r^{I}, k^{I}\right]$MDS code are merged into a single codeword of a final$\left[n^{F}=\lambda k^{I}+r^{F}, k^{F}=\lambda k^{I}\right]$MDS code. Some initial code symbols are retained in the final codeword, while others are newly generated. The access cost is the total number of symbols read or written during the conversion procedure to produce new symbols. In this paper, we present three convertible code constructions. The first two require$\lambda \leq r^{I}$and impose certain divisibility conditions on$r^{I}$and the field size$q$. However, they minimize the access cost incurred for generating each new symbol individually and for all symbols cumulatively. The third construction, a modification of an earlier construction by Kong, minimizes the cumulative access cost and works for all parameters. All the code constructions require field sizes linear in the block length of the final code and achieve the smallest known field sizes across MDS convertible codes in the literature.