Abstract
We discuss the problem of finding a minimal aliasing code among a class of systematic single-error-correction codes that are suitable to be implemented within DRAM die, as opposed to external ECC used in memory controller outside of DRAM chip. We prove a sharp lower bound of aliasing probability and propose a simple method to come up with a code that meets the bound. By an experiment, we also demonstrate that a randomly chosen code is likely to have much more aliasings with overwhelmingly high probability.
Highlights
Dynamic random access memory (DRAM) process technology has been aggressively scaled down to achieve higher density and larger bandwidth
The scaled DRAM cell is more vulnerable to data retention failure, and more closely placed DRAM cells are more likely to interfere with each other’s memory operation [1]–[8]
To deal with the reliability problem, error correction code (ECC), which is used in memory controller but outside of DRAM chip, called the external ECC in this paper, can be strengthened
Summary
Dynamic random access memory (DRAM) process technology has been aggressively scaled down to achieve higher density and larger bandwidth. It is easy to see that, for the above decoding procedure to have a SEC capability, the parity check matrix H needs to be constructed with nonzero columns to ensure that any single bit error does not lead to a zero-value syndrome, which is reserved for valid codewords. By Propositions 2 and 3, a parity check matrix H = [M | C] for a systematic SEC code is made of distinct nonzero columns with an invertible submatrix C. The matrix (b) was obtained by removing three columns from M of (a) Both the codes determined by (a) and (b) are SEC-capable and systematic because every column is distinct nonzero and its identity submatrix is invertible. It is typical to have the submatrix C as an identity matrix, we consider parity check matrices like (c) for our search for minimal aliasing codes. The rest of this section is devoted to showing that k/2 − 1 is a lower bound for the number of aliasing triples (Theorem 10), so that we can conclude that our construction gives a minimal aliasing code
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