Thoroughly wash the cucumber, radishes, and carrot, then slice them thinly into rounds or strips.
In a small saucepan, combine the rice vinegar, water, honey, apple cider vinegar, and salt. Heat over medium heat until the honey is completely dissolved.
Thinly slice the garlic and ginger, then add them to the vinegar mixture. Allow to cool completely.
In a clean, sealable jar, layer the vegetables, then pour the cooled sweet and sour vinegar mixture over them.
Use a spoon to press the vegetables down, ensuring they are completely submerged in the liquid. Seal the jar.
Refrigerate for at least 24 hours to allow the flavors to meld.
Before serving, sprinkle with toasted sesame seeds. Serve as a side dish with sushi, rice dishes, or grilled meats.
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